He turns first to the last category, eternal truths, because these are the simplest. Examples of eternal truths include the truths of mathematics and propositions such as, "It is impossible for the same thing to be and not to be at the same time" or "He who thinks cannot but exist while he thinks." These are statements of fact that, we perceive, cannot fail to be true. Though they do not have any concrete existence in the world, Descartes urges us, they certainly must be said to exist in some way.
Eternal truths are very important to Descartes' project. These are the purely intellectual ideas that he wants us to be discovering now, and they are the truths to which we all have access, so long as we withdraw from the senses. He, therefore, wants to give them some sort of real existence in the world. However, though Descartes is adamant that eternal truths exist in some way, he is not entirely clear on how, exactly, they are supposed to exist. There are several options available to him.
First, these truths might exist as their instantiations in the world. So, for instance, the truth "two plus two equals four" would exist as pairs of things in the world that together create quads. Descartes, however, would not be happy with this route. Even if there were no pairs of things in the world, Descartes would still want to say that "two plus two equals four" holds true. He would not want the existence of these truths to depend so heavily on the way the world actually is.
Another option Descartes has, and one that it often looks like he is taking, is to say that eternal truths only exist on our mind. In principle I.49 Descartes refers to them as "eternal truths which reside in our mind." This makes it sound as if eternal truths only exist insofar as someone is thinking about them. If there were no minds to believe "two plus two equals four" then there would be no such truth. Obviously, Descartes would not want this to be the case, any more than he would want the existence of eternal truths to depend on worldly instantiations. In addition, there is a second problem with this option: it makes eternal truths far too subjective. If eternal truths only exist insofar as they are in someone's mind, then whose thought matters? Do they only exist insofar as they are in my mind, in all our minds, or in God's mind? Can they exist for some people and not for others? Eternal truths are supposed to be the same for all of us, so how could they belong to our subjective minds? A final, related, problem with this view is that it makes it sound as if eternal truths are just properties of mind, since thoughts themselves are just properties of mind. Clearly, Descartes does not want eternal truths to exist as properties.
Luckily, there is a third route open to Descartes, and this is the route that he really seems to take. Eternal truths do not have any concrete existence. Instead they have a special sort of intentional existence, which is to say that they exist as the possible objects of thought. They are the things that we think about when we think about geometry, physics, mathematics, essences, etc. They do not need to actually be thought in order to exist, rather they exist as things of which can be thought.