mathematics possess no advantage over other sciences, but what they derive from a better phraseology; and that all of them might attain to the same characters of simplicity and of certainty, if we knew how to give them signs equally perfect."

Leibnitz seems to point at an idea of the same sort, in those obscure and enigmatical hints (not altogether worthy, in my opinion, of his powerful and comprehensive genius) which he has thrown out, about the miracles to be effected by a new art of his own invention; to which art he sometimes gives the name of Ars Combinatoria Characteristica, and sometimes of Ars Combinatoria Generalis ac Vera. In one of his letters to Mr. Oldenburg, he speaks of a plan he had long been meditating, of treating of the science of mind by means of mathematical demonstrations. "Many wonderful things," he adds, "of this kind have occurred to me; which, at some future period I shall explain to the public with that logical precision which the subject requires."* In the same letter, he intimates his belief in the possibility of inventing an art, "which, with an exactitude resembling that of mechanism, may render the operations of reason steady and visible, and, in their effects on the minds of others, irresistible." After which he proceeds thus:

"Our common algebra, which we justly value so highly, is no more than a branch of that general art which I have here in view. But, such as it is, it puts it out of our power to commit an error, even although we should wish to do so; while it exhibits truth to our eyes like a picture stamped on paper by means of a machine. It must at the same time be recollected, that algebra is indebted for whatever it accomplishes in the demonstration of general theorems to the suggestions of a higher science; a science which I have been accustomed to call characteristical combination; very different, however, in its nature, from that which these words are likely at first to suggest to the hearer. The marvellous utility of this art I hope to illustrate, both by precepts and examples, if I shall be so fortunate as to enjoy health and leisure.

"It is impossible for me to convey an adequate idea of it in a short description. But this I may venture to assert, that no instrument (or organ) could easily be imagined of more powerful efficacy for promoting the improvement of the human understanding; and that, supposing it to be adopted, as the common method of philosophizing, the time would very soon arrive, when we should be able to form conclusions concerning God and the mind, with not less certainty than we do at present concerning figures and numbers." (Wallisii Opera, vol. iii. p. 621.)

"Multa in hoc genere mira à me sunt observata, quæ aliquando, quo par est rigore, exposita dabo."

"Quod velut mechanica ratione fixam et visibilem et (ut ita dicam) irresistibilem reddat rationem."

The following passage is translated from another letter of Leibnitz to the same correspondent:

"The matter in question depends on another of much higher moment; I mean, on a general and true art of combination, of the extensive influence of which I do not know that any person has yet been fully aware. This, in truth, does not differ from that sublime analysis, into the recesses of which Des Cartes himself, as far as I can judge, was not able to penetrate. But, in order to carry it into execution, an alphabet of human thoughts must be previously formed and for the invention of this alphabet, an analysis of axioms is indispensably necessary. I am not, however, surprised, that nobody has yet sufficiently considered it; for we are, in general, apt to neglect what is easy; and to take many things for granted from their apparent evidence; faults which, while they remain uncorrected, will forever prevent us from reaching the summit of things intellectual, by the aid of a calculus adapted to moral as well as to intellectual science." (Wallisii Opera, p. 633.)*

In these extracts from Leibnitz, as well as in that quoted from Condillac, in the beginning of this article, the essential distinction. between mathematics and the other sciences, in point of phraseology, is entirely overlooked. In the former science, where the use of an ambiguous word is impossible, it may be easily conceived how the solution of a problem may be reduced to something resembling the operation of a mill-the conditions of the problem, when once translated from the common language into that of algebra, disappearing entirely from the view; and the subsequent process being almost mechanically regulated by general rules, till the final result is obtained. In the latter, the whole of the words about which our reasonings are conversant, admit, more or less, of different shades of meaning; and it is only by considering attentively the relation in which they stand to the immediate context, that the precise idea of the author in any particular instance is to be ascertained. In these sciences, accordingly, the constant and unremitting exercise of the attention is indispensably necessary, to prevent us, at every step of our progress, from going astray.

On this subject I have made various remarks in a volume lately published; to which I beg leave here to refer, in order to save the trouble of unnecessary repetitions. (Philosophical Essays, p. 153, et seq.) From what I have there said, I trust it appears that, in following any train of reasoning, beyond the circle of the mathematical sciences, the mind must necessarily carry on, along with

* As these reveries of this truly great man are closely connected with the subsequent history of logical speculation in more than one country of Europe, I have been induced to incorporate them, in an English version, with my own disquisitions. Some expressions, which, I am sensible, are not altogether agreeable to the idiom of our language, might have been easily avoided, if I had not felt it incumbent on me, in translating an author whose meaning, in this instance, I was able but very imperfectly to comprehend, to deviate as little as possible from his own words.

the logical deduction expressed in words, another logical process of a far nicer and more difficult nature;-that of fixing, with a rapidity which escapes our memory, the precise sense of every word which is ambiguous by the relation in which it stands to the general scope of the argument. In proportion as the language of science becomes more and more exact, the difficulty of this task will be gradually diminished; but let the improvement be carried to any conceivable extent, not one step will have been gained in accelerating that era, so sanguinely anticipated by Leibnitz and Condillac, when our reasonings in morals and politics shall resemble, in their mechanical regularity, and in their demonstrative certainty, the investigations of algebra. The improvements which language receives, in consequence of the progress of knowledge, consisting rather in a more precise distinction and classification of the various meanings of words, than in a reduction of these meanings in point of number, the task of mental induction and interpretation may be rendered more easy and unerring; but the necessity of this task can never be superseded, till every word which we employ shall be as fixed and invariable in its signification as an algebraical character or as the name of a geometrical figure.

In the mean time, the intellectual superiority of one man above another, in all the different branches of moral and political philosophy, will be found to depend chiefly on the success with which he has cultivated these silent habits of inductive interpretationmuch more, in my opinion, than on his acquaintance with those rules which form the great objects of study to the professed logician. In proof of this, it is sufficient for me to remind my readers, that the whole theory of syllogism proceeds on the supposition that the same word is always to be employed precisely in the same sense, (for otherwise, the syllogism would be vitiated by consisting of more than three terms;) and, consequently, it takes for granted, in every rule which it furnishes for the guidance of our reasoning powers, that the nicest and by far the most difficult part of the logical process has been previously brought to a successful termination.

A

In treating of a different question, I have elsewhere remarked, that although many authors have spoken of the wonderful mechanism of speech, no one has hitherto attended to the far more wonderful mechanism which it puts into action behind the scene. similar observation will be found to apply to what is commonly called the art of reasoning. The scholastic precepts which profess to teach it, reach no deeper than the very surface of the subject; being all of them confined to that part of the intellectual process which is embodied in the form of verbal propositions. On the most favorable supposition which can be formed with respect to them, they are superfluous and nugatory; but, in many cases, it is to be apprehended, that they interfere with the right conduct of the understanding by withdrawing the attention from the cultivation

of that mental logic on which the soundness of our conclusions essentially depends, and in the study of which, although some general rules may be of use, every man must be, in a great measure, his own master.*

In the practical application of the foregoing conclusions, it cannot fail to occur, as a consideration equally obvious, and important, that, in proportion as the objects of our reasoning are removed from the particular details with which our senses are conversant, the difficulty of these latent inductive processes must be increased. This is the real source of that incapacity for general speculation, which Mr. Hume has so well described as a distinguishing characteristic of uncultivated minds. "General reasonings seem intricate, merely because they are general; nor is it easy for the bulk of mankind to distinguish, in a great number of particulars, that common circumstance in which they all agree, or to extract it, pure and unmixed, from the other superfluous circumstances. Every judgment or conclusion with them is particular. They cannot enlarge their views to those universal propositions which comprehend under them an infinite number of individuals, and include a whole science in a single theorem. Their eye is confounded with such an extensive prospect, and the conclusions deduced from it, even though clearly expressed, seem intricate and obscure." (Essay on Commerce.)

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Difficult, however, and even impossible as the task of general speculation is to the bulk of mankind, it is nevertheless true, that it is the path which leads the cautious and skilful reasoner to all his most certain, as well as most valuable conclusions in morals and in politics. If a theorist, indeed, should expect, that these conclusions are in every particular instance to be realized, he would totally misapprehend their nature and application; inasmuch as they are only to be brought to an experimental test, by viewing them on an extensive scale, and continuing our observations during a long period of time. "When a man deliberates," says Mr. Hume, concerning his conduct in any particular affair, and forms schemes in politics, trade, economy, or any business in life, he never ought to draw his arguments too fine, or connect too long a chain of consequences together. Something is sure to happen that will disconcert his reasoning, and produce an event different from what he expected. But when we reason upon general subjects, one may justly affirm, that our speculations can scarcely ever be too fine, provided they be just; and that the difference between a common man and a man of genius is chiefly seen in the shallowness or depth of the principles on which they proceed." The same author afterwards excellently observes, "That general principles, however intricate they may seem, must always prevail, if they be just and

* Those who are interested in this discussion, will enter more completely into my views, if they take the trouble to combine what is here stated with some observations I have introduced in the First Part, chap. iv. sec. 2.

sound, in the general course of things, though they may fail in particular cases; and that it is the chief business of philosophers to regard the general course of things."-"I may add," continues Mr. Hume," that it is also the chief business of politicians, especially in the domestic government of the state, where the public good, which is, or ought to be, their object, depends on the concurrence of a multitude of causes; not, as in foreign politics, on accidents and chances, and the caprices of a few persons." (Essay on Commerce.)*

To these profound reflections of Mr. Hume, it may be added, although the remark does not bear directly on our present argument, that, in the systematical application of general and refined rules to their private concerns, men frequently err from calculating their measures on a scale disproportionate to the ordinary duration of human life. This is one of the many mistakes into which projectors are apt to fall and hence the ruin which so often overtakes them, while sowing the seeds of a harvest which others are to reap. A few years more might have secured to themselves the prize which they had in view; and changed the opinion of the world (which is always regulated by the accidental circumstances of failure or of success) from contempt of their folly, into admiration of their sagacity and perseverance.

It is observed by the Comte de Bussi, that "time remedies all mischances; and that men die unfortunate, only because they did not live long enough. Mareschal d'Estrée, who died rich at a hundred, would have died a beggar, had he lived only to eighty." The maxim, like most other apohthegms, is stated in terms much too unqualified; but it may furnish matter for many interesting reflections, to those who have surveyed with attention the characters which have passed before them on the stage of life; or who amuse themselves with marking the trifling and fortuitous circumstances by which the multitude are decided, in pronouncing their verdicts of foresight or of improvidence.

IV. Continuation of the Subject-Peculiar and supereminent Advantages possessed by Mathematicians, in consequence of their definite Phraseology.

If the remarks contained in the foregoing articles of this section. be just, it will follow, that the various artificial aids to our reasoning powers which have been projected by Leibnitz and others, proceed on the supposition, a supposition which is also tacitly assumed in

*This contrast between the domestic and the foreign policy of a state, occurs more than once in Mr. Hume's writings; (see in particular the first paragraphs of his Essay on the Rise of Arts and Sciences.) A similar observation had long before been made by Polybius. "There are two ways by which every kind of gov ernment is destroyed either by some accident that happens from without; or some evil that arises within itself. What the first will be, it is not always easy to foresee; but the latter is certain and determinate."-Book vi. ex. 3. (Hampton's Translation.)