6.  Hence the reality of Mathematical Knowledge

I doubt not but it will be easily granted, that the knowledge we have of mathematical truths is not only certain, but real knowledge; and not the bare empty vision of vain, insignificant chimeras of the brain:  and yet, if we will consider, we shall find that it is only of our own ideas.  The mathematician considers the truth and properties belonging to a rectangle or circle only as they are in idea in his own mind.  For it is possible he never found either of them existing mathematically, i.e. precisely true, in his life.  But yet the knowledge he has of any truths or properties belonging to a circle, or any other mathematical figure, are nevertheless true and certain, even of real things existing:  because real things are no further concerned, nor intended to be meant by any such propositions, than as things really agree to those archetypes in his mind.  Is it true of the idea of a triangle, that its three angles are equal to two right ones?  It is true also of a triangle, wherever it really exists.  Whatever other figure exists, that it is not exactly answerable to that idea of a triangle in his mind, is not at all concerned in that proposition.  And therefore he is certain all his knowledge concerning such ideas is real knowledge:  because, intending things no further than they agree with those his ideas, he is sure what he knows concerning those figures, when they have barely an Ideal existence in his mind, will hold true of them also when they have A real existance in matter:  his consideration being barely of those figures, which are the same wherever or however they exist.

7.  And of Moral.

And hence it follows that moral knowledge is as capable of real certainty as mathematics.  For certainty being but the perception of the agreement or disagreement of our ideas, and demonstration nothing but the perception of such agreement, by the intervention of other ideas or mediums; our moral ideas, as well as mathematical, being archetypes themselves, and so adequate and complete ideas; all the agreement or disagreement which we shall find in them will produce real knowledge, as well as in mathematical figures.

8.  Existence not required to make Abstract Knowledge real.

[For the attaining of knowledge and certainty, it is requisite that we have determined ideas:] and, to make our knowledge real, it is requisite that the ideas answer their archetypes.  Nor let it be wondered, that I place the certainty of our knowledge in the consideration of our ideas, with so little care and regard (as it may seem) to the real existence of things:  since most of those discourses which take up the thoughts and engage the disputes of those who pretend to make it their business to inquire after truth and certainty, will, I presume, upon examination, be found to be general propositions, and notions in which existence