thing, that every process of reasoning is perfectly analogous to an algebraical operation. What I mean by "the other intellectual processes distinct from reasoning, which it is necessary for us sometimes to employ in the investigation of truth," will, I hope, appear clearly from the following remarks.

Mental operations subsidiary to reasoning. In algebraical investigations, it is well known, that the practical application of a general expression is frequently limited by the conditions which the hypothesis involves; and that, in consequence of a want of attention to this circumstance, some mathematicians of the first eminence have been led to adopt the most paradoxical and absurd conclusions. Without this cautious exercise of the judgment in the interpretation of the algebraical language, no dexterity in the use of the calculus will be sufficient to preserve us from error. Even in algebra, therefore, there is an application of the intellectual powers perfectly distinct from any process of reasoning, and which is absolutely necessary for conducting us to the truth.

In Geometry, we are not liable to adopt the same paradoxical conclusions, as in algebra; because the diagrams, to which our attention is directed, serve as a continual check on our reasoning powers. These diagrams exhibit, to our very senses, a variety of relations among the quantities under consideration, which the language of algebra is too general to express; in consequence of which, we are not conscious of any effort of the judgment distinct from a process of reasoning. As every geometrical investigation, however, may be expressed algebraically, it is manifest, that in geometry, as well as in algebra, there is an exercise of the intellectual powers, distinct from the logical process; although, in the former science, it is rendered so easy, by the use of diagrams, as to escape our attention.

The same source of error and of absurdity, which exists in algebra, is to be found, in a much greater degree, in the other branches of knowledge. Abstracting entirely from the ambiguity of language, and supposing also our reasonings to be logically accurate, it would still be necessary for us, from time to time, in all our speculations, to lay aside the use of words, and to

have recourse to particular examples or illustrations, in order to correct and to limit our general conclusions. To a want of attention to this circumstance, a number of the speculative absurdities which are current in the world might, I am persuaded, be easily traced.

Besides, however, this source of error, which is in some degree common to all the sciences, there is a great variety of others, from which mathematics are entirely exempted; and which perpetually tend to lead us astray in our philosophical inquiries. Of these, the most important is, that ambiguity in the signification of words, which renders it so difficult to avoid employing the same expressions in different senses, in the course of the same process of reasoning. This source of mistake, indeed, is apt, in a much greater degree, to affect our conclusions in metaphysics, morals, and politics, than in the different branches of natural philosophy; but if we except mathematics, there is no science whatever, in which it has not a very sensible influence. In algebra, we may proceed with perfect safety through the longest investigations, without carrying our attention beyond the signs, till we arrive at the last result. But in the other sciences, excepting in those cases in which we have fixed the meaning of all our terms by accurate definitions, and have rendered the use of these terms perfectly familiar to us by very long habit, it is but seldom that we can proceed in this manner, without danger of error. In many cases, it is necessary for us to keep up, during the whole of our investigations, a scrupulous and constant attention to the signification of our expressions; and in most cases, this caution in the use of words is a much more difficult effort of the mind, than the logical process. But still this furnishes no exception to the general doctrine already delivered; for the attention we find it necessary to give to the import of our words arises only from the accidental circumstance of their ambiguity, and has no essential connection with that process of the mind which is properly called reasoning; and which consists in the inference of a conclusion from premises. In all the sciences, this process of the mind is perfectly analogous to an algebraical operation; or, in other words, (when the

meaning of our expressions is once fixed by definitions,) it may be carried on entirely by the use of signs, without attending, during the time of the process, to the things signified.

The conclusion to which the foregoing observations lead, appears to me to be decisive of the question, with respect to the objects of our thoughts when we employ general terms; for if it be granted, that words, even when employed without any reference to their particular signification, form an instrument of thought sufficient for all the purposes of reasoning; the only shadow of an argument in proof of the common doctrine on the subject, (I mean that which is founded on the impossibility of explaining this process of the mind on any other hypothesis,) falls to the ground. Nothing less, surely, than a conviction of this impossibility, could have so long reconciled philosophers to an hypothesis unsupported by any direct evidence; and acknowledged, even by its warmest defenders, to involve much difficulty and mystery.

Uses of illustrated and abstract reasoning. It does not fall within my plan to enter, in this part of my work, into a particular consideration of the practical consequences which follow from the foregoing doctrine. I cannot, however, help remarking the importance of cultivating, on the one hand, a talent for ready and various illustration; and, on the other, a habit of reasoning by means of general terms. The former talent is necessary, not only for correcting and limiting our general conclusions, but for enabling us to apply our knowledge, when occasion requires, to its real practical use. The latter serves the double purpose, of preventing our attention from being distracted during the course of our reasonings, by ideas which are foreign to the point in question, and of diverting the attention from those conceptions of particular objects and particular events which might disturb the judgment, by the ideas and feelings which are apt to be associated with them, in consequence of our own casual experience.

This last observation points out to us, also, one principal foundation of the art of the orator. As his object is not so much to inform and satisfy the understandings of his hearers, as to

force their immediate assent; it is frequently of use to him to clothe his reasonings in that specific and figurative language, which may either awaken in their minds associations favorable to his purpose, or may divert their attention from a logical examination of his argument. A process of reasoning so expressed, affords at once an exercise to the judgment, to the imagination, and to the passions; and is apt, even when loose and inconsequential, to impose on the best understandings.

It appears further, from the remarks which have been made, that the perfection of philosophical language, considered either as an instrument of thought, or as a medium of communication with others, consists in the use of expressions, which, from their generality, have no tendency to awaken the powers of conception and imagination; or, in other words, it consists in its approaching, as nearly as possible, in its nature, to the language of algebra. And hence the effects which long habits of philosophical speculation have in weakening, by disuse, those faculties of the mind, which are necessary for the exertions of the poet and the orator; and of gradually forming a style of composition, which they who read merely for amusement, are apt to censure for a want of vivacity and of ornament.*

III. Remarks on the opinions of some modern philosophers on the subject of the foregoing section. -After the death of Abelard, through whose abilities and eloquence the sect of Nominalists had enjoyed, for a few years, a very splendid triumph, the system of the Realists began to revive; and it was soon so completely reëstablished in the schools, as to prevail, with little or no opposition, till the fourteenth century. What the circum

* "Language, like light, is a medium and the true philosophical style, like light from a north window, exhibits objects clearly and distinctly without soliciting attention to itself. In painting subjects of amusement indeed, language may gild somewhat more, and color with the dyes of fancy; but where information is of more importance than entertainment, though you cannot throw too strong a light, you should carefully avoid a colored one. The style of some writers resembles a bright light placed between the eye, and the thing to be looked at. The light shows itself, and hides the object." — Gilpin.

stances were, which led philosophers to abandon a doctrine, which seems so strongly to recommend itself by its simplicity, it is not very easy to conceive. Probably the heretical opinions, which had subjected both Abelard and Roscelinus to the censure of the Church, might create a prejudice also against their philosophical principles; and probably, too, the manner in which these principles were stated and defended was not the clearest, nor the most satisfactory. The principal cause, however, I am disposed to think, of the decline of the sect of Nominalists, was their want of some palpable example, by means of which they might illustrate their doctrine. It is by the use which algebraists make of the letters of the alphabet in carrying on their operations, that Leibnitz and Berkeley have been most successful in explaining the use of language as an instrument of thought: and, as in the twelfth century, the algebraical art was entirely unknown, Roscelinus and Abelard must have been reduced to the necessity of conveying their leading idea by general circumlocutions; and must have found considerable difficulty in stating it in a manner satisfactory to themselves; a consideration, which, if it accounts for the slow progress which this doctrine made in the world, places in the more striking light the genius of those men whose sagacity led them, under so great disadvantages, to approach to a conclusion so just and philosophical in itself, and so opposite to the prevailing opinions of their age.

In the fourteenth century, this sect seems to have been almost completely extinct; their doctrine being equally reprobated by the two great parties which then divided the schools, the followers of Duns Scotus and of Thomas Aquinas. These, although they differed in their manner of explaining the nature of universals, and opposed each other's opinions with much asperity, yet united in rejecting the doctrine of the Nominalists, not only as absurd, but as leading to the most dangerous consequences. At last, William Occam, a native of England, and a scholar of Duns Scotus, revived the ancient controversy, and, with equal ability and success, vindicated the long-abandoned philosophy of Roscelinus. From this time the dispute was car