ment or difagreement it feeks with the next intermediate idea, which it ufes as a proof; for if it were not fo, that yet would need a proof; fince without the perception of fuch agreement or difagreement, 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 fome intervening idea, as a common measure to show their agreement or disagreement: By which it is plain that every step in reafoning 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 difagreement of the ideas, concerning which we inquire, vifible and certain. So that to make any thing a demonftration, it is neceflary to perceive the immediate agreement of the intervening ideas, whereby the agreement or difagreement 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 progreflion of the demonftration, must also be carried exactly in the mind, and a man must be fure that no part is left out; which, because in long deductions, and the use of many proofs, the memory does not always fo readily and exactly retain; therefore it comes to pafs, that this is more imperfect than intuitive. knowledge, and men embrace often falfehood for de-. monstrations.

§ 8. Hence the mistake, Ex præcognitis et præconceffis.. THE neceffity of this intuitive knowledge, in each step. of fcientifical or demonftrative reafoning, gave occafion, imagine, to that mistaken axiom, that all reafoning was ex præcognitis et præconceffis; which how far it is mistaken, I fhall have occafion to fhow more at large, when I come to confider propofitions, and particularly thofe propofitions which are called max-. ims; and to fhow that it is by a mistake, that they are fuppofed to be the foundation of all our know.. ledge and reasonings.

9. Demonftration not limited to Quantity. IT has been generally taken for granted, that mathematics alone are capable of demonftrative certainty; but to have fuch an agreement or difagreement as may intuitively be perceived, being, as I imagine, not the privilege of the ideas of number, extenfion and figure alone; it may poffibly be the want of due method and application in us, and not of fufficient evidence in things, that demonftration has been thought to have fo little to do in other parts of knowledge, and been scarce fo much as aimed at by any but mathematicians. For whatever ideas we have, wherein the mind can perceive the immediate agreement or difagreement that is between them, there the mind is capable of intuitive knowledge; and where it can perceive the agreement or difagreement of any two ideas, by an intuitive perception of the agreement or difagreement they have with any intermediate ideas, there the mind is capable of demonstration, which is not limited to ideas of extenfion, figure, number, and their modes.

10. Why it has been so thought.

THE reason why it has been generally fought for, and fuppofed to be only in thofe, I imagine has been not only the general ufefulness of thofe fciences, but becaufe, in comparing their equality or excefs, the modes of numbers have every the least difference very clear and perceivable; and though in extenfion, every the leaft excefs is not fo perceptible, yet the mind has found out ways to examine and difcover demonftratively the juft equality of two angles, or extenfions, or figures; and both these, i. e. numbers and figures, can be fet down by vifible and lasting marks, wherein the ideas under confideration are perfectly determined; which for the most part they are not, where they are marked only by names and words.

§ II.

Bur in other fimple ideas, whofe modes and differences are made and counted by degrees, and not quantity, we have not so nice and accurate a distinction of

their differences, as to perceive or find ways to meafure their juft equality, or the leaft differences. For thofe other fimple ideas, being appearances or fenfations, produced in us by the fize, figure, number and motion of minute corpufcles fingly infenfible, their different degrees alfo depend upon the variation of fome, or all of those causes; which fince it cannot be obferved by us in particles of matter, whereof each is too fubtle to be perceived, it is impoffible for us to have any exact measures of the different degrees of thefe fimple ideas. For fuppofing the fenfation of idea we name whitenefs be produced in us by a certain number of globules, which having a verticity about their own centres, ftrike upon the retina of the eye, with a certain degree of rotation, as well as progreffive fwiftnefs, it will hence eafily follow, that the more the fuperficial parts of any body are fo ordered, as to reflect the greater number of globules of light, and to give them that proper rotation which is fit to produce this fenfation of white in us, the more white will that body appear, that from an equal space fends to the retina the greater number of fuch corpufcles, with that peculiar fort of motion. I do not fay, that the nature of light confifts in very small round globules, nor of whitenefs in fuch a texture of parts, as gives a certain rotation to thefe globules when it reflects them; for I am not now treating phyfically of light or colours: But this, I think, I may fay, that I cannot (and I would be glad any one would make intelligible that he did) conceive how bodies without us can any ways affect our fenfes, but by the immediate contact of the fentible bodies themfelves, as in tafting and feeling, or the impulfe of fome infenfible particles coming from them. as in feeing, hearing, and fmelling; by the different impulfe of which parts, caufed by their different fize. figure and motion, the variety of fenfations is produced in us.

$12.

WHETHER then they be globules, or no; or whether they have a verticity about their own centres, that

produce the idea of whiteness in us, this is certain, that the more particles of light are reflected from a body, fitted to give them that peculiar motion which produces the fenfation of whitenefs in us, and poffibly too, the quicker that peculiar motion is, the whiter does the body appear, from which the greater number are reflected, as is evident in the fame piece of paper put in the fun-beams, in the fhade, and in a dark hole; in each of which it will produce in us the idea of whitenefs in far different degrees.

§ 13. Why it has been fo thought.

NoT knowing therefore what number of particles, nor what motion of them is fit to produce any precife degree of whiteness, we cannot demonftrate the certain equality of any two degrees of whiteness, becaufe we have no certain ftandard to measure them by, nor means to diftinguish every the leaft real difference, the only help we have being from our senses, which in this point fail us. But where the difference is fo great, as to produce in the mind clearly distinct ideas, whofe differences can be perfectly retained, there thefe ideas of colours, as we fee in different kinds, as blue and red, are as capable of demonftration, as ideas of number and extenfion. What I have here said of whiteness and colours, I think, holds true in all fecondary qualities, and their modes.

§14. Senfitive Knowledge of particular exiftence. THESE two, viz. intuition and demonftration, are the degrees of our knowledge; whatever comes fhort of one of thefe, with what affurance foever embraced, is but faith, or opinion, but not knowledge, at leaft in all general truths. There is, indeed, another perception of the mind, employed about the particular existence of finite beings without us, which going beyond bare probability, and yet not reaching perfectly to either of the foregoing degrees of certainty, paffes under the name of knowledge. There can be nothing more certain, than that the idea we receive from an external object is in our minds; this is intuitive knowledge. But whether there be any thing more than

barely that idea in our minds, whether we can thence certainly infer the existence of any thing without us, which corresponds to that idea, is that, whereof some men think there may be a queftion made; because men may have fuch ideas in their minds, when no fuch thing exists, no fuch object affects their fenfes. But yet here, I think, we are provided with an evidence, that puts us paft doubting; for I ask any one, whether he be not invincibly confcious to himself of a different perception, when he looks on the fun by day, and thinks on it by night; when he actually taftes wormwood, or smells a rofe, or only thinks on that favour or odour? We as plainly find the difference there is between any idea revived in our minds by our own memory, and actually coming into our minds by our fenfes, as we do between any two diftinct ideas. If any one fay, a dream may do the fame thing, and all these ideas may be produced in us without any external objects, he may pleafe to dream that I make him this anfwer: 1. That it is no great matter, whether I remove his fcruple, or no; where all is but a dream, reafoning and arguments are of no use, truth and knowledge nothing. 2. That I believe he will allow a very manifeft difference between dreaming of being in the fire, and being actually in it. But yet if he be resolved to appear fo fceptical, as to maintai that what I call being actually in the fire is nothing but a dream, and that we cannot thereby certainly know, that any such thing as fire actually exists without us: I answer, that we certainly finding that pleafure or pain follows upon the application of certain objects to us, whofe exiftence we perceive, or dream that we perceive by our fenfes; this certainty is as great as our happiness or mifery, beyond which we have no concernment to know, or to be. So that I think, we may add to the two former forts of knowledge this alfo, of the existence of particular external objects, by that perception and confcioufnefs we have of the actual entrance of ideas from them, and allow these three degrees of knowledge, viz. intuitive, de