are equal, is of relation: iron is susceptible of magnetical impressions, is, of co-existence: God is, is of real existence. Though identity and co-existence are truly nothing, but relations, yet they are so peculiar ways of agreement, or disagreement, of our ideas, that they deserve well to be considered as distinct heads, and not under relation in general; since they are so different grounds of affirmation and negation, as will easily appear to any one who will but reflect on what is said in several places of this essay. I should not proceed to examine the several degrees of our knowledge, but that it is necessary first to consider the different acceptations of the word knowledge.

§8. Knowledge actual or habitual.-There are several ways wherein the mind is possessed of truth; each of which is called knowledge.

First, There is actual knowledge, which is the present view the mind has of the agreement or disagreement of any of its ideas, or of the relation they have one to another.

Secondly, A man is said to know any proposition, which having been once laid before his thoughts, he evidently perceived the agreement or disagreement of the ideas whereof it consists; and so lodged it in his memory, that whenever that proposition comes again to be reflected on, he, without doubt or hesitation, embraces the right side, assents to, and is certain of, the truth of it. This, I think, one may call habitual knowledge; and thus a man may be said to know all those truths, which are lodged in his memory by a foregoing clear and full perception, whereof the mind is assured past doubt, as often as it has occasion to reflect on them. For our finite understandings being able to think clearly and distinctly but on one thing at once, if men had no knowledge of any more than what they actually thought on, they would all be very ignorant: and he that knew most, would know but one truth, that being all he was able to think on at one time.

§9. Habitual knowledge two-fold.—Of habitual knowledge, there are also, vulgarly speaking, two degrees:

First, The one is of such truths laid up in the memory, as whenever they occur to the mind, it actually perceives the relation is between those ideas. And this is in all those truths, whereof we have an intuitive knowledge, where the ideas themselves, by an immediate view, discover their agreement or disagreement one with another.

Secondly, The other is of such truths, whereof the mind having been convinced, it retains the memory of the conviction, without the proofs. Thus a man that remembers certainly, that he once perceived the demonstration, that the three angles of a triangle are equal to two right ones, is certain that he knows it, because he cannot doubt the truth of it. In his adherence to a truth, where the demonstration, by which it was at first known, is forgot, though a man may be thought rather to believe his memory, than really to know, and this way of entertaining a truth seemed formerly to me like something between opinion and knowledge, a sort of assurance which exceeds bare belief, for that relies on the testimony of another; yet upon a due examination, I find it comes not short of perfect certainty, and is in effect true knowledge. That which is apt to mislead our first thoughts into a mistake in this matter, is, that the agreement or disagreement of the ideas in this case is

not perceived, as it was at first, by an actual view of all the intermediate ideas, whereby the agreement or disagreement of those in the proposition was at first perceived; but by other intermediate ideas, that shew the agreement or disagreement of the ideas contained in the proposition whose certainty we remember. For example, in this proposition, that the three angles of a triangle are equal to two right ones, one who has seen and clearly perceived the demonstration of this truth, knows it to be true, when that demonstration is gone out of his mind; so that at present it is not actually in view, and possibly cannot be recollected; but he knows it in a different way from what he did before. The agreement of the two ideas joined in that proposition, is perceived, but it is by the intervention of other ideas than those which at first produced that perception. He remembers, i. e. he knows (for remembrance is but the reviving of some past knowledge), that he was once certain of the truth of this proposition, that the three angles of a triangle are equal to two right ones. The immutability of the same relations between the same immutable things, is now the idea that shews him, that if the three angles of a triangle were once equal to two right ones, they will always be equal to right ones. And hence he comes to be certain, that what was once true in the case, is always true; what ideas once agreed, will always agree; and consequently what he once knew to be true, he will always know to be true, as long as he can remember that he once knew it. Upon this ground it is, that particular demonstrations in mathematics afford general knowledge. If then the perception that the same ideas will eternally have the same habitudes and relations, be not a sufficient ground of knowledge, there could be no knowledge of general propositions in mathematics; for no mathematical demonstration would be any other than particular: and when a man had demonstrated any proposition concerning one triangle or circle, his knowledge would not reach beyond that particular diagram. If he would extend it farther, he must renew his demonstration in another instance, before he could know it to be true in another like triangle, and so on; by which means, one could never come to the knowledge of any general propositions. Nobody, I think, can deny that Mr. Newton certainly knows any proposition, that he now at any time reads in his book, to be true, though he has not in actual view that admirable chain of intermediate ideas, whereby he at first discovered it to be true. Such a memory as that, able to retain such a train of particulars, may be well thought beyond the reach of human faculties. When the very discovery, perception, and laying together that wonderful connexion of ideas, is found to surpass most readers' comprehension. But yet it is evident the author himself knows the proposition to be true, remembering he once saw the connexion of those ideas, as certainly as he knows such a man wounded another, remembering that he saw him run through. But because the memory is not always so clear as actual perception, and does in all men more or less decay in length of time, this, amongst other differences, is one, which shews, that demonstrative knowledge is much more imperfect than intuitive, as we shall see in the following chapter.

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CHAP. II.

OF THE DEGREES OF OUR KNOWLEDGE.

§ 1. Intuitive. All our knowledge consisting, as I have said, in the view the mind has of its own ideas, which is the utmost light and greatest certainty, we with our faculties, and in our way of knowledge, are capable of, it may not be amiss to consider a little the degrees of its evidence. The different clearness of our knowledge seems to me to lie in the different way of perception the mind has of the agreement or disagreement of any of its ideas. For if we will reflect on our own ways of thinking, we shall find, that sometimes the mind perceives the agreement or disagreement of two ideas immediately by themselves, without the intervention of any other: and this, I think, we may call intuitive knowledge. For in this, the mind is at no pains in proving or examining, but perceives the truth, as the eye doth light, only by being directed towards it. Thus the mind perceives that white is not black, that a circle is not a triangle, that three are more than two, and equal to one and two. Such kind of truths the mind perceives at the first sight of the ideas together, by bare intuition, without the intervention of any other idea; and this kind of knowledge is the clearest, and most certain, that human frailty is capable of. This part of knowledge is irresistible, and like bright sunshine, forces itself immediately to be perceived, as soon as ever the mind turns its view that way; and leaves no room for hesitation, doubt, or examination, but the mind is presently filled with the clear light of it. It is on this intuition, that depends all the certainty and evidence of all our knowledge, which certainty every one finds to be so great, that he cannot imagine, and therefore not require, a greater; for a man cannot conceive himself capable of a greater certainty, than to know that any idea in his mind is such as he perceives it to be; and that two ideas, wherein he perceives a difference, are different, and not precisely the same. He that demands a greater certainty than this, demands he knows not what, and shews only that he has a mind to be a sceptic, without being able to be so. Certainty depends so wholly on this intuition, than in the next degree of knowledge, which I call demonstrative, this intuition is necessary in all the connexions of the intermediate ideas, without which, we cannot attain knowledge and certainty.

§ 2. Demonstrative. The next degree of knowledge is where the mind perceives the agreement or disagreement of any ideas, but not immediately. Though wherever the mind perceives the agreement or disagreement of any of its ideas, there be certain knowledge; yet it does not always happen, that the mind sees that agreement or disagreement, which there is between them, even where it is discoverable; and in that case, remains in ignorance, and at most, gets no farther than a probable conjecture. The reason why the mind cannot always perceive presently the agreement or disagreement of two ideas, is because those ideas concerning whose agreement or disagreement the inquiry is made, cannot by the mind be so put together, as to shew it. In this case

then, when the mind cannot so bring its ideas together, as by their immediate comparison, and, as it were, juxta-position, or application one to another, to perceive their agreement or disagreement, it is fain, by the intervention of other ideas (one or more, as it happens), to discover the agreement or disagreement which it searches; and this is that which we call reasoning. Thus the mind being willing to know the agreement or disagreement in bigness, between the three angles of a triangle, and two right ones, cannot by an immediate view and comparing them, do it; because the three angles of a triangle cannot be brought at once, and be compared with any one or two angles; and so of this the mind has no immediate, no intuitive, knowledge. In this case, the mind is fain to find out some other angles, to which the three angles of a triangle have an equality; and finding those equal to two right ones, comes to know their equality to two right ones.

§ 3. Depends on proofs.-Those intervening ideas, which serve to shew the agreement of any two others, are called proofs; and where the agreement or disagreement is by this means plainly and clearly perceived, it is called demonstration, it being shewn to the understanding, and the mind made to see that it is so. A quickness in the mind to find out these intermediate ideas (that shall discover the agreement or disagreement of any other), and to apply them right, is, I suppose, that which is called sagacity.

§ 4. But not so easy. This knowledge by intervening proofs, though it be certain, yet the evidence of it is not altogether so clear and bright, nor the assent so ready, as an intuitive knowledge. For though in demonstration, the mind does at last perceive the agreement or disagreement of the ideas it considers, yet it is not without pains and attention; there must be more than one transient view to find it. A steady application and pursuit are required to this discovery; and there must be a progression by steps and degrees, before the mind can in this way arrive at certainty, and come to perceive the agreement or repugnancy between two ideas that need proofs, and the use of reason to shew it.

§ 5. Not without precedent.-Another difference between intuitive and demonstrative knowledge, is, that though in the latter all doubt be removed, when, by the intervention of the intermediate ideas, the agreement or disagreement is perceived; yet before the demonstration there was a doubt, which, in intuitive knowledge, cannot happen to the mind that has its faculty of perception left to a degree capable of distinct ideas, no more than it can be a doubt to the eye (that can distinctly see white and black), whether this ink and this paper be all of a colour. If there be sight in the eyes, it will at first glimpse, without hesitation, perceive the words printed on this paper, different from the colour of the paper; and so if the mind have the faculty of distinct perceptions, it will perceive the agreement or disagreement of those ideas that produce intuitive knowledge. If the eyes have lost the faculty of seeing, or the mind of perceiving, we in vain inquire after the quickness of sight in one, or clearness of perception in the other.

§ 6. Not so clear.-It is true, the perception produced by demonstration, is also very clear; yet it is often with a great abatement of that evident lustre and full assurance, that always accompany that which I

call intuitive, like a face reflected by several mirrors one to another, where, as long as it retains the similitude and agreement with the object, it produces a knowledge; but it is still in every successive reflection with a lessening of that perfect clearness and distinctness, which is in the first; till at last, after many removes, it has a great mixture of dimness, and is not at first sight so knowable, especially to weak eyes. Thus it is with knowledge, made out by a long train of proofs.

§ 7. Each step must have intuitive evidence.-Now, in every step reason makes in demonstrative knowledge, there is an intuitive knowledge of that agreement or disagreement, it seeks with the next intermediate idea, which it uses as a proof: for if it were not so, that yet would need a proof; since without the perception of such agreement or disagreement, there is no knowledge produced. If it be perceived by itself, it is intuitive knowledge; if it cannot be perceived by itself, there is need of some intervening idea, as a common measure to shew their agreement or disagreement. By which it is plain, that every step in reasoning, that produces knowledge, has intuitive certainty: which when the mind perceives, there is no more required, but to remember it, to make the agreement or disagreement of the ideas, concerning which we inquire, visible and certain. So that to make any thing a demonstration, it is necessary to perceive the immediate agreement of the intervening ideas, whereby the agreement or disagreement of the two ideas under examination (whereof the one is always the first, and the other the last, in the account) is found. This intuitive perception of the agreement or disagreement of the intermediate ideas, in each step and progression of the demonstration, must also be carried exactly in the mind, and a man must be sure that no part is left out; which, because in long deductions, and the use of many proofs, the memory does not always so readily and exactly retain; therefore it comes to pass, that this is more imperfect than intuitive knowledge, and men embrace often falsehood for demonstrations.

§ 8. Hence the mistake, ex præcognitis et præconcessis. — The necessity of this intuitive knowledge, in each step of scientifical or demonstrative reasoning, gave occasion, I imagine, to that mistaken axiom, that all reasoning was ex præcognitis et præconcessis; which how far it is mistaken, I shall have occasion to shew more at large, when I come to consider propositions, and particularly those propositions which are called maxims; and to shew that it is by a mistake, that they are supposed to be the foundations of all our knowledge and reasonings.

§ 9. Demonstration not limited to quantity.-It has been generally taken for granted, that mathematics alone are capable of demonstrative certainty; but to have such an agreement or disagreement, as may intuitively be perceived, being, as I imagine, not the privilege of the ideas of number, extension, and figure alone, it may possibly be the want of due method and application in us, and not of sufficient evidence in things, that demonstration has been thought to have so little to do in other parts of knowledge, and been scarce so much as aimed at by any but mathematicians. For whatever ideas we have, wherein