Getting the best out of a book-reading exercise A doctorate involves plumbing the depths of a problem and often this involves a lot of knowledge breadth-wise in related areas. Though the latter is provided by coursework, education about a precise "tool" that a doctorate may need during solving his problem is unlikely to have been provided by the courses. Often the PhD may run into huge tomes of material which he has been introduced to by Wikipedia. Intuitively he/she may feel that somewhere in those volumes lies a theorem or an idea which can provide vital keys to solving his own problem. So I come to my questions:


What should a student do to get the best out of a book-reading exercise? The question in turn assumes the student has arrived by some means at the best book for serving his purpose.
There is a trade-off involved in reading books tangential to your field: you may spend a lot of time groping in the dark looking for a bounty which may never be there. How does one gauge the potential applicability of a book to his work? When does a student decide to pull the plug on such an effort?


Related Question: Skimming through a paper
First, I would like to comment on some of your sentences. Then I will answer to your 2 questions.


Often the PhD may run into huge tomes of material which he has been introduced to by Wikipedia: I am not sure that I understand this sentence well. Are you using Wikipedia for bootstrapping and then proceed from page to page in wikipedia, hoping to find something useful, or do you use another source of idea, but jump to wikipedia to understand in a faster way the entities/concepts you find in the first place ? BTW, in most cases I am not sure Wikipedia is something we should use at the research level. The quality is clearly increasing, but this is far from perfect, and some essential intuitions are missing sometimes (almost always ?).
Intuitively he/she may feel that somewhere in those volumes lies a theorem or an idea which can provide vital keys to solving his own problem. This sentence makes me think that you maybe focus too much on a specific problem.


Now, my answers :


When I read a paper or a book, I proceed the same way: First I quickly go through the whole thing (for a book, the whole thing is generally a chapter). Then I try to understand the key intuitions and results. At that point I don't try to understand how they are proven. Then I switch to other things. When I am thinking of a problem, or when I am on other concepts, sometimes a flash occurs and I have the intuition that something I have read can be useful, then I go back to the paper/book and try to totally understand the result. Here, this means more or less homework like in the old days ;)
Reading tangential materials is a necessity. I proceed as stated above, and to make sure that I will not spend all my time on this, I decided to always spend the same amount of time reading on other fields. For me this is roughly 1 day a week, which is 1/3 to 1/2 of the time I can use for research.