at the square of the
hypothenuse is equal to the square of the two sides_, is a
proposition which expresses a relation between these two figures.
_That three times five is equal to the half of thirty_, expresses a
relation between these numbers. Propositions of this kind are
discoverable by the mere operation of thought without dependence on
whatever is anywhere existent in the universe. Though there never
were a circle or a triangle in nature, the truths demonstrated by
Euclid would for ever retain their certainty and evidence.

"Matters of fact, which are the second objects of human reason, are
not ascertained in the same manner, nor is an evidence of their
truth, however great, of a like nature with the foregoing. The
contrary of every matter of fact is still possible, because it can
never imply a contradiction, and is conceived by the mind with the
same facility and distinctness, as if ever so conformable to
reality. _That the sun will not rise to-morrow_, is no less
intelligible a proposition, and implies no more contradiction, than
the affirmation, _that it will rise_. We should in vain, therefore,
attempt to demonstrate its falsehood. Were it demonstratively
false, it would imply a contradiction, and could never be
distinctly conceived by the mind."--(IV. pp. 32, 33.)

The distinction here drawn between the truths of geometry and other
kinds of truth is far less sharply indicated in the _Treatise_, but as
Hume expressly disowns any opinions on these matters but such as are
expressed in the _Inquiry_, we may confine ourselves to the latter; and
it is needful to look narrowly into the propositions here laid down, as
much stress has been laid upon Hume's admission that the truths of
mathematics are intuitively and demonstratively certain; in other
words, that they are necessary and, in that respect, differ from all
other kinds of belief.

What is meant by the assertion that "propositions of this kind are
discoverable by the mere operation of thought without dependence on what
is anywhere existent in the universe"?

Suppose that there were no such things as impressions of sight and touch
anywhere in the universe, what idea could we have even of a straight
line, much less of a triangle and of the relations between its sides?
The fundamental proposition of all Hume's philosophy is that ideas are
copied from impressio