of some of the most important theorems in the Differential Calculus and in Finite Differences. Boole's application of this principle to Formal Logic was bold and original in the highest degree. Having shown that any system of propositions may be represented by equations involving symbols, x, y, &c., which, whenever interpretation is possible, are subject to laws identical in form with the laws of a system of quantitative symbols, susceptible only of the values 0 and 1, he proceeds :

But as the formal processes of reasoning depend only upon the laws of the symbols, and not upon the nature of their interpretation, we are permitted to treat the above symbols, x, y, z, as if they were quantitative symbols of the kind above described. We may, in fact, lay aside the logical interpretation of the symbols in the given equation; convert them into quantitative symbols, susceptible only of the values 0 and 1; perform upon them, as such, all the requisite processes of solution; and, finally, restore to them their logical interpretation. [And this is the mode of procedure which Boole actually adopts.] The processes to which the symbols x, y, z, regarded as quantitative, and of the species above described, are subject, are not limited by those conditions of thought to which they would, if performed upon purely logical symbols, be subject, and a freedom of operation is given to us in the use of them without which the inquiry after a general method in logic would be a hopeless quest.'

Numerous applications of his method are given by Boole in the Laws of Thought.' That method has for its object the 'determination of any element in any proposition, however 'complex, as a logical function of the remaining elements. 'Instead of confining our attention to the "subject," and "predicate," regarded as simple terms, we can take any 'element, or any combination of elements entering into either 'of them; make that element, or that combination, the "sub'ject" of a new proposition; and determine what its "predicate" shall be, in accordance with the data afforded to us.' In this way, also, any system of equations whatever, by which propositions or combinations of propositions, can be represented, may be analysed, and all the conclusion' which those propositions involve, be deduced from them. In the light of this method, Boole examines the Aristotelian logic and some of its modern extensions. He shows that conversion, syllogism, &c., are not the ultimate processes of logic, but themselves rest upon and are resolvable into, ulterior and more elementary processes. And the conclusion at which he arrives with respect to the nature and extent of the scholastic logic is, that it is not a 'science, but a collection of scientific truths, too incomplete to

De Morgan and Jevons on Boole's System.

179

'form a system of themselves, and not sufficiently fundamental 'to serve as the foundation upon which a perfect system may

'rest.'

It ought perhaps to be distinctly stated here that Boole did not propose his calculus of 0 and I as a substitute for common reasoning. He was well aware that any, the most perfect, system of formal logic must possess a theoretical rather than a practical importance. The perfection of the method of logic,' he says, 'may be chiefly valuable as an evidence of the specu'lative truth of its principles. To supersede the employment of 'common reasoning, or to subject it to the rigour of technical forms, would be the last desire of one who knows the value of 'that intellectual toil and warfare which imparts to the mind an athletic vigour, and teaches it to contend with difficulties, and to rely upon itself in emergencies. Nevertheless,' he adds, 'cases may arise in which the value of a scientific pro'cedure, even in those things which fall confessedly under the ordinary dominion of the reason, may be felt and acknowledged.'

The power of the method is most strikingly exemplified in its application to the theory of probabilities, but that is a branch of the subject on which it is impossible here to enter. We could not exhibit the formula for the expansion, elimination, &c., of logical functions, or show how such formulæ may be applied to the analysis of propositions, without covering our pages with symbols that would render them as unintelligible and uninviting to the general reader as a work written in Arabic. We must therefore content ourselves with having briefly stated the axiomatic laws on which Boole's system is based, and pointed out their formal connection with the fundamental laws of Algebra. 'Mr. Boole's generalisation of the forms of logic,' says Professor De Morgan, 'cannot be sepa'rated from mathematics, since it not only demands Algebra, 'but such taste for thought about the notation of Algebra as is rarely acquired without much and deep practice. When the 'ideas thrown out by Mr. Boole shall have borne their full 'fruit, Algebra, although only founded on ideas of number in the first instance, will appear like a sectional model of the 'whole form of thought. Its forms considered apart from their ' matter, will be seen to contain all the forms of thought in 'general. The anti-mathematical logician says that it makes thought a branch of Algebra, instead of Algebra a branch of 'thought. It makes nothing; it finds; and it finds the laws of 'thought symbolised in the forms of Algebra.'-English Cyclopædia, Art. Logic.

In a very ingenious little work, entitled 'Pure Logic, or the Logic of Quality apart from Quantity,' Mr. W. Stanley Jevons has lately developed a system of deductive reasoning closely analogous to, and in some respects identical with, that given by Boole. Of the merits or defects of Mr. Jevons' system we shall not now speak, but we cannot close this article without saying a word or two touching his objections to Boole's method. Those objections are entitled to attention if for no other reason than this, that they evidently proceed from one who has made a careful study of the work which he undertakes to criticise. Yet we are far from being satisfied that they are well founded. His first objection is, that Boole's symbols are essentially dif'ferent from the names or symbols of common discourse.' Here the question turns wholly upon the office assigned in the Laws of Thought to the symbol +, which is there used to connect terms which are mutually exclusive. It is objected that in common discourse the conjunctions and,' 'or,' are not invariably so used. This, however, is distinctly admitted by Boole (chap. IV., § 6), who nevertheless vindicates his mode of using the symbol +, and, as it appears to us, upon good and sufficient grounds. Mr. Jevons' second objection, viz., that there are no such operations as addition and subtraction in pure logic,' is founded upon a misapprehension. The mental operation indicated in Boole's system by the sign +, is that by which from the conception of two distinct classes of things, we form the conception of that group or collection of things which those classes taken together compose. Now this, as a mental operation, is wholly different from that process of the mind by which we pass from the arithmetical notion of one object to that of two, three, or more of the same. In like manner logical subtraction, expressed by the sign, and which is the opposite or negative of logical addition, is entirely distinct as a mental operation from arithmetical subtraction. Yet Mr. Jevons enters into a somewhat elaborate argument to prove that arithmetical addition and subtraction have no place in pure logic! Another objection, viz., that Boole's system is inconsistent with the self-evident law of thought, the law of unity (A + A = A),' is equally unfounded. Mr. Jevons attaches his own meaning to the symbol +, and this is essentially distinct and different from that assigned to the symbol by Boole. These, therefore, in the two systems, are not one and the same symbol; they are two, and any argument built upon their assumed identity must of course be fallacious. In Boole's system, the expression A+ A, is not equivalent to A, neither does this expression in general admit of interpretation. We cannot conceive of the addition of a class A to itself.

Mr. Jevons' fourth and last objection relates to Boole's interpretations of the numerical symbols 0, 1, &c. Those interpretations are necessitated by the nature of the notation adopted. In every symbolical system there are concessions which must be made to notation.

Mr. Jevons rejects the calculus of 0 and 1, and proposes in its place a method which he thinks is equally powerful, and at the same time more simple, intelligible, and purely logical. His little book may be read with advantage in connection with the 'Laws of Thought.' In his general estimate of Boole's system we entirely concur. 'It is not to be denied,' he says, 'that 'Boole's system is complete and perfect within itself. It is, perhaps, one of the most marvellous and admirable pieces of reasoning ever put together. Indeed, if Professor Ferrier, in his "Institutes of Metaphysics," is right in holding that the 'chief excellence of a system is in being reasoned and consis'tent within itself, then Professor Boole's is nearly or quite the 'most perfect system ever struck out by a single writer."

We understand that Boole has left behind him a considerable quantity of logical manuscripts, and that these are to be published either in a separate form, or in a new edition of the 'Laws of Thought.' His works are his noblest monument, but his friends and admirers are raising other memorials. Of these we may mention in particular, a memorial window in the cathedral at Lincoln, and another in the College Hall at Cork, the glass alone of the latter window is to cost £350. His widow has placed a mural tablet to his memory in the church of Ballintemple, the inscription on which is as follows:-"To the memory ' of George Boole, D.C.L., F.R.S., First Professor of Mathematics in the Queen's College, Cork, in whom the highest order of Intellect, cultivated by unwearied Industry, pro'duced the Fruits of deep Humility and Child-like Trust. He

was born in Lincoln, on the 2nd November, 1815. And died ' at Ballintemple on the 8th December, 1864. "For ever, O Lord, thy Word is settled in Heaven."

There are some men whose office gives them celebrity; there are other men who give celebrity to their office. Boole was one of the latter: The Chair of Mathematics which he filled at Cork would not have made his name illustrious; but that chair has, through the genius and labours of its first occupant, acquired a reputation which only powers of the highest order in his successors can sustain and perpetuate.

R. H.

ART. VII. (1.) A Bill to amend the Representation of the People in Parliament in England and Wales. 6th June, 1866.

(2.) Electoral Returns, 1865-1866.

(3.) The New Reform Bill. By R. DUDLEY BAXTER, M.A.

(4.) The History of the Reform Bill. By the Rev. W. N. MOLESWORTH, M.A.

A veteran politician discoursing lately on Reform, drew attention to the great difference between the year 1832 and the year 1866. In 1832,' he said, 'all was feeling, and passion, and declamation. In 1866, all is argument and calculation.'

At first sight this may seem a doubtful aphorism. The late debates in the House of Commons have not been so very calm. Argument there has been, and calculation in abundance, but still more feeling, quite as much passion, and not a little declamation. No one listening to the organized clamour of the Tory Mountain, no one who heard the well-concerted laughter of the young Tory roughs below the gangway, or witnessed their persistent and passionate intemperance as the discussion rose in importance, would at the time be disposed to think that passion had forsaken the question of Reform. As for declamation, the parliamentary hero of the hour, the member for Calne himself, has proved himself to be, not indeed the first statesman, but beyond all competition, the first declaimer of the day. From first to last his speeches on the subject of Reform, plausible in the abstract, brilliant beyond eulogy as epidictic displays, have only served to betray a long fit of intellectual passion. If judged by the standard of practical statesmanship, Mr. Lowe's speeches must be deemed very inapplicable to the actual facts and premises of the problem, peculiarly local and English, with which, as an English statesman, he pretended to deal, while, in reality, he was dissecting an ideal problem, from the point of view of a French fanatic and political idealist. There is hardly a fallacy, hardly a vaticination of ills to arise from Reform uttered thirty years ago during the great Reform debates of 1832, which, in some form or other, Mr. Lowe has not recast and employed against the Reform Bill of 1866. And yet, as we must admit, so many men inside the House, and even outside, were found to sympathise with his passionate and intemperate prejudices, that Mr. Lowe's reputation at this moment in Parliament as an orator and statesman is probably second to that of no other man in the kingdom. So far then it would seem as if the anti-reform passions of 1866 were not very far behind those of 1832.

But although this may be true of the House of Commons,