least as far as they apply to myself, before entering on any new speculations, concerning our reasoning powers; and shall, at the same time, introduce some occasional illustrations of the principles which I formerly endeavoured to establish.

To prevent the possibility of misrepresentation, I state Dr. Reid's objection in his own words.

"Berkeley, in his reasoning against abstract general ideas, seems "unwillingly or unwarily to grant all that is necessary to support "abstract and general conceptions.

"A man (says Berkeley) may consider a figure merely as trian"guiar, without attending to the particular qualities of the angles, (6 or relations of the sides. So far he may abstract. But this will 66 never prove that he can frame an abstract general inconsistent "idea of a triangle."

Upon this passage Dr. Reid makes the following remark: "If a man may consider a figure merely as triangular, he must have some conception of this object of his consideration; for no man can consider a thing which he does not conceive. He has a conception, therefore, of a triangular figure, merely as such. I know no more that is meant "by an abstract general conception of a triangle."

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"He that considers a figure merely as triangular (continues the "same author) must undersand what is meant by the word trian"gular. If to the conception he joins to this word, he adds any "particular quality of angles or relation of sides, he misunderstands "it, and does not consider the figure merely as triangular. Whence "I think it is evident, that he who considers a figure merely as "triangular, must have the conception of a triangle, abstracting "from any quality of angles or relations of sides."*

For what appears to myself to be a satisfactory answer to this reasoning, I have only to refer to the first volume of these Elements. The remarks to which I allude are to be found in the third section of chapter fourth ;† and I must beg leave to recommend them to the attention of my readers, as a necessary preparation for the following discussion.

In the farther prosecution of the same argument, Dr. Reid lays hold of an acknowledgment which Berkeley has made, "That we 66 may consider Peter so far forth as man, or so far forth as animal, "in as much as all that is perceived is not considered."—" It may "here (says Reid) be observed, that he who considers Peter so far "forth as man, or so far forth as animal, must conceive the meaning "of those abstract general words man and animal; and he who "conceives the meaning of them, has an abstract general conception."

According to the definition of the word conception, which I have given in treating of that faculty of the mind, a general conception is an obvious impossibility. But, as Dr. Reid has chosen to annex a more extensive meaning to the term than seems to me consistent with precision, I would be far from being understood to object to P. 195, 3d edit.

* Reid's Intellectual Powers, p. 483, 4to edit.

his conclusion, merely because it is inconsistent with an arbitrary definition of my own. Let us consider, therefore, how far his doctrine is consistent with itself; or rather, since both parties are evidently so nearly agreed about the principal fact, which of the two have adopted the more perspicuous and philosophical mode of stating it.

In the first place, then, let it be remembered as a thing admitted on both sides, "that we have a power of reasoning concerning a "figure considered merely as triangular, without attending to the "particular qualities of the angles, or relations of the sides;" and also, 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 circum"stances abstracted, as separate objects of conception. But this is "by no means requisite in every case in which abstraction is sub"servient 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 con"cerning extension and figure, without any reference to colour, "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 colour 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."* 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 reason, 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 circumstances 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 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

"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 appellatious. 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 com pare and find, that the three angles of that triangle, are equal to those right angles that stand by it. But if another triangle be shewn him, different in shape from the former, he cannot know, without a new labour, whether the three angles of that also be equal to the same. But he that bath 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, andthe angles three; and that that was all for which he named it a triangle; will holdly conclude universally, that such equality of an gles 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 labour 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 I. chap. iv.

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 perfectly 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

* 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 familiarize the mind of his pupil with the fundamental principles of the science, as a preparation for the study of physics and other parts of mixed mathematics, it cannot be denied, that such a practice would be far less favourable 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 seen to threaten; in as much as the student, without any reflection whatever on logical principles, generali. zes 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 impres sive 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 errours into which the judgment might be led by the specific peculiarities of the example. Hence, by the way, a strong argument in favour of the prac tice recommended by Bacon, of connecting emblems with praenotions, 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 imagi nation. 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 conclusiong to others. But the prosecution of these hints would lead me too far astray from the sub. ject of this section.

the question as the breadth or the colour 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 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 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 labour 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 possi"ble cases of a problem at one view; and often in one general theo

rem comprehending whole sciences; which deduced at length "into propositions, and demonstrated after the manner of the an"cients, might well become the subject of large treatises."* Of this remark, Halley gives an instance in a formula, which, when he

* Philos. Transact. No. 205. Miscell. Cur. Vol. I. p. 348.