way arrive at certainty, and come to perceive the agreement or repugnancy between two ideas that need proofs and the use of reason to show it.

§ 5. Not without precedent Doubt.

ANOTHER difference between intuitive and demonftrative knowledge, is, that though in the latter all doubt be removed, when by the intervention of the intermediate ideas the agreement or difagreement is perceived; yet before the demonftration 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 diftinct ideas, no more than it can be a doubt to the eye (that can diftinctly fee white and black) whether this ink and this paper be all of a colour. If there be fight in the eyes, it will at first glimpfe, without hesitation perceive the words printed on this paper different from the colour of the paper; and fo if the mind have the faculty of diftinct perceptions, it will perceive the agreement or difagreement of thofe ideas that produce intuitive knowledge. If the eyes have loft the faculty of feeing, or the mind of perceiving, we in vain inquire after the quickness of fight in one, or clearness of perception in the other. § 6. Not fo clear.

It is true, the perception produced by demonftration is also very clear, yet it is often with a great abatement of that evident luftre and full affurance, that always accompany that which I call intuitive; like a face reflected by feveral mirrors one to another, where, as long as it retains the fimilitude and agreement with the object, it produces a knowledge; but it is still in every fucceffive reflection with a leffening of that perfect clearness and diftinctnefs, which is in the first, till at laft, after many removes, it has a great mixture of dimness, and is not at first fight fo 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 reafon makes in demonftrative knowledge, there is an intuitive knowledge of that agree

ment or difagreement it feeks with the next intermediate idea, which it ufes as a proof; for if it were not so, 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 fhow 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 laft in the account) is found. This intuitive perception of the agreement or difagreement of the intermediate ideas, in each step and progreffion of the demonftration, must also be carried exacly 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 demonstrations.

§8. Hence the mistake, Ex præcognitis et præconceffis.. THE neceffity of this intuitive knowledge, in each step of fcientifical or demonftrative reafoning, gave occa fion, I imagine, to that mistaken axiom, that all reafoning was ex præcognitis et præconceffis; which how far it is mistaken, I fhall have occafion to show more at large, when I come to confider propofitions, and particularly those propofitions which are called max. ims; and to show 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 demonstration 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 per ceive 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 demonftration, which is not limited to ideas of extenfion, figure, number, and their modes.

10. Why it has been fo 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 usefulness of thofe fciences, but be cause, in comparing their equality or excefs, the modes of numbers have every the leaft difference very clear and perceivable; and though in extenfion, every the leaft excess is not fo perceptible, yet the mind has found out ways to examine and difcover demonftratively the just equality of two angles, or extenfions, or figures; and both thefe, i. e. numbers and figures, can be fet down by visible and lafting 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.

BUT in other fimple ideas, whofe modes and differences are made and counted by degrees, and not quantity, we have not fo 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 sensa tions, 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 thofe caufes; 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 whiteness 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 swiftnefs, 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 glo. bules, 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 fenfible 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 fensations is produced in us.

12.

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

Book IV. 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, because 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 fenfes, 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 demonstration, as ideas of number and extenfion. What I have here faid 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 least in all general truths. There is, indeed, another perception of the mind, employed about the particular exiftence 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