It sounds like the character is trying to find a spot, and he thinks he is sitting near the spot. "As best he could determine" means that based on all the information he has, he believes he is near the spot. Let's think of another example. Let's imagine there's a rock collector who finds a rock that he has never seen before. He looks in a book to identify it. In the book, he finds a rock called "agatite," and the picture and description match the rock that he found. He says: "As best I can determine, this rock is agatite." He means that he is not absolutely certain, but based on the information that he has, he believes the rock is agatite. In everyday speech, people are more likely to say "as best I can tell." The meaning is the same, just more casual-sounding. Example: "As best I can tell, he was lying." (I am not certain that he was lying, but all the evidence I have suggests that he was lying.) This phrase "as best [person] [can/could] [determine/tell]" is unique. Besides "determine" or "tell," other verbs aren't used very often. However, there's a slight variation: "As best as [person] [can/could]." Sometimes there's a verb after "can" or "could." Examples: (Tom is listening to Bob play the guitar, and Bob is playing poorly.) Tom: That's bad! Why don't you play better than that? Bob: I'm playing as best as I can. Bob is saying that he can't play the guitar better than how he is playing it. (Jessica and Florence are arranging flowers.) Jessica: That doesn't look very good. Florence: That's as best as I can arrange them. Florence is saying that she can't arrange the flowers better than how she arranged them. Do you understand? If you still have questions I can try to explain it some more.

Maybe somebody else can explain this more easily. But I will do my best :) "As best he could ..." means the best he could do; to the best of his ability, or his best effort. So you could say, "The old lady climbed the stairs as best she could." which means she did the best climbing that she could (but probably not a very good job). "As best he could determine" means that, given the information he has available, this is his "best guess". Well, not really a "guess", but "conclude" if you know that word. For example, "I have determined that you are in love with me!" means that I have thought about everything, and decided that you are in love with me. Please note that "I have determined ...", and "As best he could determine ..." is formal language that you may read and write, but you probably won't hear in conversation. Unless the person is very educated :) If the word "determined" is confusing you -- look up the word "ascertain" in an English dictionary, it might help you. The meanings are very similar. But be careful because "determined" has 2 different meanings ... Sorry about this messy answer! Anyway, it's the same meaning as: "As far as he could work out, he was sitting near the spot." Some examples: "He determined that the killer was John!" (This means, he investigated, considered the evidence, and finally came to the conclusion that John was the killer) "As best as I can determine, she doesn't work on Mondays." (This means, "I'm pretty sure she doesn't work on Mondays", or "It would appear that she doesn't work on Mondays, based on my observations" ... The same meaning.) "The location of the new sports arena is still undetermined." (This means, the location is still undecided. Wow this answer is really messy, I hope somebody can give a better one for you ...

Thanks :-)

Taking a little piece from both answers below (by Graham and Ian) A more "casual" and common way to say it is "As far as I can tell..." Though in this case it would be "As far as he could tell..."