You can't have both, Hobbes!

Either pick geometry or pick human inability to conceive the infinite.

Throughout part 1 of Leviathan, Hobbes seeks to clarify his work by defining the elements of human nature. In doing so, he concludes that knowledge is broken down into two types. The simple type refers to, "knowledge of fact." This is anything that can be empirically verified. As Hobbes puts it, "nothing else but sense and memory." On the other side of knowledge, Hobbes postulates, "knowledge of the consequence of one affirmation to another." To further define the second type of knowledge, Hobbes uses an example that relates to geometry. The example he gives is, "if the figure shown be a circle, then any straight line through the center shall it into two equal parts." Whats troubling about this is that he goes on to define this as a science, particularly one that pertains to the philosopher. This is troubling because this seems like rationality to me, which is not a kind of knowledge. Hobbes' second type of knowledge can only be a working of or a faculty of the mind rather than a piece of knowledge that is given. In other words, I see this type of reason not as some sort of presented fact, but as process by which we use logic to assess the given situation. I take Hobbes' account of knowledge to be wrong in this case.
This also makes me think of mathematical proofs. If we try to prove the previous example, one will quickly come to the most basic axioms - facts that must be accepted as true even though they have no empirical grounds whatsoever (cant measure the two halves of the circle or even make a straight line or circle, etc...) Unless Hobbes gives a fuller account of knowledge by grounding these basic axioms, we might as well consider his proof of knowledge to be, "we know because we know." It sounds silly when put this way but, nonetheless, he has not sufficiently established the grounds upon which he can say we know things intellectually. One more thought - does the quote, "There is no idea or conception of anything we call infinite" interfere with his second type of knowledge? The consideration of geometry made me consider that. How would we have knowledge that the two sections, when divided by a straight line are equal. Either we cant assume that, or we have some conception of the infinite.