libly
certain, that all the knowledge we attain concerning these ideas is
real, and reaches things themselves. Because in all our thoughts,
reasonings, and discourses of this kind, we intend things no further
than as they are conformable to our ideas. So that in these we cannot
miss of a certain and undoubted reality.

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 r