David Hume, answered from the texts and cited to the page.
The division you are asking about runs through everything I wrote on the scope and limits of human understanding. All objects of human reason or enquiry, I argued, fall naturally into one of two kinds: Relations of Ideas and Matters of Fact.1 Relations of ideas — the province of geometry, algebra, arithmetic — are propositions discoverable by the mere operation of thought, without dependence on what is anywhere existent in the universe.2
Their denials imply contradiction; their truth is either intuitive or demonstrative. Euclid's theorems would retain their certainty even if no circle or triangle had ever existed in nature.3 Matters of fact are entirely otherwise. The contrary of any matter of fact remains always possible; it implies no contradiction and is conceived by the mind with the same facility as the fact itself.4
That the sun will not rise tomorrow is no less intelligible a proposition than that it will — and we should in vain attempt to demonstrate its falsehood by reason alone.5 The fork's cutting edge appears when you ask what lies outside these two categories. The Treatise presses the same division into moral philosophy directly: if the understanding operates only by comparing ideas or inferring matters of fact, and virtue is to be discovered by the understanding, then virtue must be an object of one of these two operations — there is no third.6
The rationalist who claims morality is susceptible of demonstration must show that vice and virtue consist in relations among ideas; but the four relations that admit of certainty — resemblance, contrariety, degrees in quality, and proportions in quantity or number — belong as properly to inanimate matter as to human actions, which makes them useless as the ground of moral distinction.7
The fork is therefore not merely a classification of propositions. It is a diagnostic instrument. Apply it to any claim that presents itself with philosophical authority, and ask: is this a relation of ideas, true by the mere operation of thought? Or is it a matter of fact, grounded in experience and memory? If it answers to neither description, you have grounds to suspect the claim carries less content than its language suggests.
All the objects of human reason or enquiry may naturally be divided into two kinds, to wit, Relations of Ideas, and Matters of Fact.An Enquiry Concerning Human Understanding, p. 12
Propositions of this kind are discoverable by the mere operation of thought, without dependence on what is anywhere existent in the universe.An Enquiry Concerning Human Understanding, p. 12
Though there never were a circle or triangle in nature, the truths demonstrated by Euclid would for ever retain their certainty and evidence.An Enquiry Concerning Human Understanding, p. 12
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.An Enquiry Concerning Human Understanding, p. 12
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.An Enquiry Concerning Human Understanding, p. 12
As the operations of human understanding divide themselves into two kinds, the comparing of ideas, and the inferring of matter of fact; were virtue discover'd by the understanding; it must be an object of one of these operations, nor is there any third operation of the understanding, which can discover it.A Treatise of Human Nature, p. 298
Resemblance, contrariety, degrees in quality, and proportions in quantity and number; all these relations belong as properly to matter, as to our actions, passions, and volitions. 'Tis unquestionable, therefore, that morality lies not in any of these relations, nor the sense of it in their discovery.A Treatise of Human Nature, p. 298