that "we may reason concerning Peter or John, considered so far forth as man, or so far forth as animal." About these facts there is but one opinion; and the only question is, Whether it throws additional light on the subject, to tell us, in scholastic language, that "we are enabled to carry on these general reasonings, in consequence of the power which the mind has of forming abstract general conceptions." To myself it appears, that this last statement (even on the supposition that the word conception is to be understood agreeably to Dr. Reid's own explanation,) can serve no other purpose than that of involving a plain and simple truth in obscurity and mystery. If it be used in the sense in which I have invariably employed it in this work, the proposition is altogether absurd and incomprehensible.

For the more complete illustration of this point, I must here recur to a distinction formerly made between the abstractions which are subservient to reasoning, and those which are subservient to imagination." In every instance in which imagination is employed in forming new wholes, by decompounding and combining the perceptions of sense, it is evidently necessary that the poet or the painter should be able to state or represent to himself the circumstances abstracted, as separate objects of conception. But this is by no means requisite in every case in which abstraction is subservient to the power of reasoning; for it frequently happens, that we can reason concerning the quality or property of an object abstracted from the rest, while, at the same time, we find it impossible to conceive it separately. Thus, I can reason concerning extension and figure, without any reference to color, although it may be doubted, if a person possessed of sight can make extension and figure steady objects of conception, without connecting with them the idea of one color or another. Nor is this always owing (as it is in the instance just mentioned) merely to the association of ideas; for there are cases, in which we can reason concerning things separately, which it is impossible for us to suppose any mind so constituted as to conceive apart. Thus we can reason concerning length, abstracted from any other dimension; although, surely, no understanding can make length, without breadth, an object of conception." (First Part, page 100.) In like manner, while I am studying Euclid's demonstration of the equality of the three angles of a triangle to two right angles, I find no difficulty in following his train of reasoning, although it has no reference whatever to the specific size or to the specific form of the diagram before me. I abstract, therefore, in this instance, from both of these circumstances presented to my senses by the immediate objects of my perceptions; and yet, it is manifestly impracticable for me either to delineate on paper, or to conceive in the mind, such a figure as shall not include the circumstance from which I abstract, as well as those on which the demonstration hinges.

In order to form a precise notion of the manner in which this

process of the mind is carried on, it is necessary to attend to the close and inseparable connexion which exists between the faculty of general reasoning, and the use of artificial language. It is in consequence of the aids which this lends to our natural faculties, that we are furnished with a class of signs, expressive of all the circumstances which we wish our reasonings to comprehend; and, at the same time, exclusive of all those which we wish to leave out of consideration. The word triangle, for instance, when used without any additional epithet, confines the attention to the three angles and three sides of the figure before us; and reminds us, as we proceed, that no step of our deduction is to turn on any one of the specific varieties which that figure may exhibit. The notion, however, which we annex to the word triangle, while we are reading the demonstration, is not the less a particular notion, that this word, from its partial or abstracted import, is equally applicable to an infinite variety of other individuals.*

These observations lead, in my opinion, to so easy an explanation of the transition from particular to general reasoning, that I shall make no apology for prosecuting the subject a little farther, before leaving this branch of my argument.

It will not, I apprehend, be denied, that when a learner first enters on the study of geometry, he considers the diagrams before him as individual objects, and as individual objects alone. In reading, for example, the demonstration just referred to, of the equality of the three angles of every triangle to two right angles, he thinks only of the triangle which is presented to him on the margin of the page. Nay, so completely does this particular figure engross his attention, that it is not without some difficulty he, in the first instance, transfers the demonstration to another triangle whose form is very different, or even to the same triangle placed in an inverted position. It is in order to correct this natural bias of the mind, that a judicious teacher, after satisfying himself that the student comprehends per

"By this imposition of names, some of larger, some of stricter signification, we turn the reckoning of the consequences of things imagined in the mind, into a reckoning of the consequences of appellations. For example, a man that hath no use of speech at all (such as is born and remains perfectly deaf and dumb) if he set before his eyes a triangle, and by it two right angles (such as are the corners of a square figure) he may by meditation compare and find, that the three angles of that triangle are equal to those right angles that stand by it. But if another triangle be shown him, different in shape from the former, he cannot know, without a new labor, whether the three angles of that also be equal to the same. But he that hath the use of words, when he observes that such equality was consequent, not to the length of the sides, nor to any particular thing in this triangle; but only to this, that the sides were straight, and the angles three; and that that was all for which he named it a triangle; will boldly conclude universally, that such equality of angles is in all triangles whatsoever; and register his invention in these general terms. Every triangle hath its three angles equal to two right angles. And thus the consequence found in one particular, comes to be registered and remembered as an universal rule; and discharges our mental reckoning of time and place; and delivers us from all labor of the mind, saving the first and makes that which was found true here, and now, to be true in all times and places."-Hobbes, Of Man, Part 1. chap. iv.

fectly the force of the demonstration, as applicable to the particular triangle which Euclid has selected, is led to vary the diagram in different ways, with a view to show him, that the very same demonstration, expressed in the very same form of words, is equally applicable to them all. In this manner he comes, by slow degrees, to comprehend the nature of general reasoning, establishing insensibly in his mind this fundamental logical principle, that when the enunciation of a mathematical proposition involves only a certain portion of the attributes of the diagram which is employed to illustrate it, the same proposition must hold true of any other diagram involving the same attributes, how much soever distinguished from it by other specific peculiarities.*

Of all the generalizations in geometry, there are none into which the mind enters so easily, as those which relate to diversities in point of size or magnitude. Even in reading the very first demonstrations of Euclid, the learner almost immediately sees, that the scale on which the diagram is constructed, is as completely out of the question as the breadth or the color of the lines which it presents to his external senses. The demonstration, for example, of the fourth proposition, is transferred, without any conscious process of reflection, from the two triangles on the margin of the page, to those comparatively large ones which a public teacher exhibits on his board or slate to a hundred spectators. I have frequently,

In order to impress the mind still more forcibly with the same conviction, some have supposed that it might be useful, in an elementary work, such as that of Euclid, to omit the diagrams altogether, leaving the student to delineate them for himself, agreeably to the terms of the enunciation and of the construction. And were the study of geometry to be regarded merely as subservient to that of logic, much might be alleged in confirmation of this idea. Where, however, it is the main purpose of the teacher (as almost always happens) to familiarise the mind of his pupil with the fundamental principles of the science, as a preparation for the study of physics and of the other parts of mixed mathematics, it cannot be denied, that such a practice would be far less favorable to the memory than the plan which Euclid has adopted, of annexing to each theorem an appropriate diagram, with which the general truth comes very soon to be strongly associated. Nor is this circumstance found to be attended in practice with the inconvenience it may seem to threaten; inasmuch as the student, without any reflection whatever on logical principles, generalizes the particular example, according to the different cases which may occur, as easily and unconsciously as he could have applied to these cases the general enunciation.

The same remark may be extended to the other departments of our knowledge in all of which it will be found useful to associate with every important general conclusion, some particular example or illustration, calculated, as much as possible, to present an impressive image to the power of conception. By this means, while the example gives us a firmer hold, and a readier command of the general theorem, the theorem, in its turn, serves to correct the errors into which the judgment might be led by the specific peculiarities of the example. Hence, by the way, a strong argument in favor of the practice recommended by Bacon, of connecting emblems with prænotions, as the most powerful of all adminicles to the faculty of memory; and hence the aid which this faculty may be expected to receive, in point of promptitude, if not of correctness, from a lively imagination. Nor is it the least advantage of this practice, that it supplies us at all times with ready and apposite illustrations to facilitate the communication of our general conclusions to others. But the prosecution of these hints would lead me too far astray from the subject of this section.

however, observed in beginners, while employed in copying such elementary diagrams, a disposition to make the copy, as nearly as possible, both in size and figure, a fac simile of the original.

The generalizations which extend to varieties of form and of position, are accomplished much more slowly; and, for this obvious reason, that these varieties are more strongly marked and discriminated from one another, as objects of vision and of conception. How difficult (comparatively speaking) in such instances, the generalizing process is, appears manifestly from the embarrassment which students experience, in applying the fourth proposition to the demonstration of the fifth. The inverted position, and the partial coincidence of the two little triangles below the base, seem to render their mutual relation so different from that of the two separate triangles which had been previously familiarized to the eye, that it is not surprising this step of the reasoning should be followed, by the mere novice, with some degree of doubt and hesitation. Indeed, where nothing of this sort is manifested, I should be more inclined to ascribe the apparent quickness of his apprehension to a retentive memory, seconded by implicit faith in his instructor; than to regard it as a promising symptom of mathematical genius.

Another, and perhaps a better, illustration of that natural logic which is exemplified in the generalization of mathematical reasonings, may be derived from those instances where the same demonstration applies, in the same words, to what are called, in geometry, the different cases of a proposition. In the commencement of our studies, we read the demonstration over and over, applying it successively to the different diagrams; and it is not without some wonder we discover, that it is equally adapted to them all. In process of time, we learn that this labor is superfluous; and if we find it satisfactory in one of the cases, can anticipate with confidence the justness of the general conclusion, or the modifications which will be necessary to accommodate it to the different forms of which the hypothesis may admit.

The algebraical calculus, however, when applied to geometry, places the foregoing doctrine in a point of view still more striking; representing," to borrow the words of Dr. Halley, "all the possible cases of a problem at one view; and often in one general theorem comprehending whole sciences; which deduced at length into propositions, and demonstrated after the manner of the ancients, might well become the subject of large treatises. (Philos. Transact. No. 205. Miscell. Cur. Vol. i. p. 348.) Of this remark, Halley gives an instance in a formula, which when he first published it, was justly regarded "as a notable instance of the great use and comprehensiveness of algebraic solutions." I allude to his formula for finding universally the foci of optic lenses; an example which I purposely select, as it cannot fail to be familiarly known to all who have the slightest tincture of mathematical and physical science.

In such instances as these, it will not surely be supposed, that while we read the geometrical demonstration, or follow the successive steps of the algebraical process, our general conceptions embrace all the various possible cases to which our reasonings extend. So very different is the fact, that the wide grasp of the conclusion is discovered only by a sort of subsequent induction; and, till habit has familiarized us with similar discoveries, they never fail to be attended with a certain degree of unexpected delight. Dr. Halley seems to have felt this strongly, when the optical formula, already mentioned, first presented itself to his mind."

In the foregoing remarks, I have borrowed my examples from mathematics, because, at the period of life when we enter on this study, the mind has arrived at a sufficient degree of maturity to be able to reflect accurately on every step of its own progress; whereas, in those general conclusions to which we have been habituated from childhood, it is quite impossible for us to ascertain, by any direct examination, what the processes of thought were, which originally led us to adopt them. In this point of view, the first doubtful and unassured steps of the young geometer, present to the logician a peculiarly interesting and instructive class of phenomena, for illustrating the growth and development of our reasoning powers. The true theory, more especially of general reasoning, may be here distinctly traced by every attentive observer; and may hence be confidently applied (under due limitations) to all the other departments of human knowledge.*

The view of general reasoning which is given above, appears to myself to af ford (without any comment) a satisfactory answer to the following argument of the late worthy and learned Dr. Price: "That the universality consists in the idea, and not merely in the name, as used to signify a number of particulars, resembling that which is the immediate object of reflection, is plain; because, was the idea to which the name answers, and which it recalls into the mind, only a particular one, we could not know to what other ideas to apply it, or what particular objects had the resemblance necessary to bring them within the meaning of the name. A person, in reading over a mathematical demonstration, certainly is conscious that it relates to some what else, than just that precise figure presented to him in the diagram. But if he knows not what else, of what use can the demonstration be to him? How is his knowledge enlarged by it? Or how shall he know afterwards to what to apply to it?"

In a note upon this passage, Dr. Price observes, that, "according to Dr. Cudworth, abstract ideas are implied in the cognoscitive power of the mind; which, he says, contains in itself virtually (as the future plant or tree is contained in the seed) general notions or exemplars of all things, which are exerted by it, or unfold and discover themselves, as occasions invite, and proper circumstances occur." 'This, no doubt," Dr. Price adds, "many will very freely condemn as whimsical and extravagant. I have, I own, a different opinion of it; but yet I should not care to be obliged to defend it."-Review of the Principal Questions in Morals, pp. 38, 39, 2nd edit.

For my own part, I have no scruple to say, that I consider this fancy of Cudworth as not only whimsical and extravagant, but as altogether unintelligible; and yet it appears to me, that some confused analogy of the same sort must exist in the mind of every person who imagines that he has the power of forming general conceptions without the intermediation of language.

In the continuation of the same note, Dr. Price seems disposed to sanction another remark of Dr. Cudworth: in which he pronounces the opinion of the Nominalists to be so ridiculous and false, as to deserve no confutation. I suspect,