15.  The next is got by Reasoning.

But though we have, here and there, a little of this clear light, some sparks of bright knowledge, yet the greatest part of our ideas are such, that we cannot discern their agreement or disagreement by an immediate comparing them.  And in all these we have need of reasoning, and must, by discourse and inference, make our discoveries.  Now of these there are two sorts, which I shall take the liberty to mention here again:—­

First, through Reasonings that are Demonstrative.

First, Those whose agreement or disagreement, though it cannot be seen by an immediate putting them together, yet may be examined by the intervention of other ideas which can be compared with them.  In this case, when the agreement or disagreement of the intermediate idea, on both sides, with those which we would compare, is plainly discerned:  there it amounts to demonstration whereby knowledge is produced, which, though it be certain, yet it is not so easy, nor altogether so clear as intuitive knowledge.  Because in that there is barely one simple intuition, wherein there is no room for any the least mistake or doubt:  the truth is seen all perfectly at once.  In demonstration, it is true, there is intuition too, but not altogether at once; for there must be a remembrance of the intuition of the agreement of the medium, or intermediate idea, with that we compared it with before, when we compare it with the other:  and where there be many mediums, there the danger of the mistake is the greater.  For each agreement or disagreement of the ideas must be observed and seen in each step of the whole train, and retained in the memory, just as it is; and the mind must be sure that no part of what is necessary to make up the demonstration is omitted or overlooked.  This makes some demonstrations long and perplexed, and too hard for those who have not strength of parts distinctly to perceive, and exactly carry so many particulars orderly in their heads.  And even those who are able to master such intricate speculations, are fain sometimes to go over them again, and there is need of more than one review before they can arrive at certainty.  But yet where the mind clearly retains the intuition it had of the agreement of any idea with another, and that with a third, and that with a fourth, &c., there the agreement of the first and the fourth is a demonstration, and produces certain knowledge; which may be called rational knowledge, as the other is intuitive.

16.  Secondly, to supply the narrowness of Demonstrative and Intuitive Knowledge we have nothing but Judgment upon probable reasoning.