under which similar things, but not their opposites, fall, and to recognize similarities and differences. This is the intellect’s comparative and analogical reasoning. When internal or external sensation operates alone, it apprehends the existence of a particular thing. When abstract thought operates alone, it knows the universals conceived within it, which may or may not exist outside the mind. Whether their individual instances exist or not can be known only through internal or external sensation.

For when you say, “It is a fact that one hundred is one-tenth of one thousand,” you are not making a judgment about anything outside the mind. Indeed, even if nothing in the world could be counted in hundreds or thousands, you would still know that the hundred conceived in your mind is one-tenth of the thousand. But when you perceive men, animals, gold, and silver, and ascertain—through your own senses or the report of someone who has perceived them—that there are one hundred men or dirhams in one place and one thousand in another, or the like, you judge that one of the two quantities counted is one-tenth of the other.

The things counted can be apprehended only through sensory perception, whereas abstract number is apprehended by the heart. Through the heart’s understanding and sensory perception together, both number and the things counted are known. Geometrical magnitudes likewise belong to this category. Thus, primary, self-evident, purely intellectual knowledge concerns only things conceived in the mind, such as number and magnitude, not things actually existing outside it.