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I am not a member of an academic institution right now, but am working on a paper for publication, partly to improve my prospects for admission to PhD programs after a long gap in my education. Two absolutely necessary books for my research are only available in academic libraries (one is also for sale on AbeBooks for about $450).

So my question is, would it be appropriate to email a former professor and ask him to get these books through an inter-library loan (ILL) for me so that I could skim them for several hours, take necessary notes, and return them on the same day?

This professor has taught me in the same field, and was personally supportive. Aside from former professors, I don't know anyone else in an academic institution, and am not in a position to fly across the country to visit libraries where these items are held.

I don't see an issue with this. In fact, you can probably do it yourself. Many libraries will offer library privileges to people for research. I live in a small town with a nice library. My local librarian has, on occasion, gotten such resources for me through contacts with a university library.

Libraries generally are there to help. They need some assurance that you will treat such privileges properly, of course, but you might just go ask to see what is available to you.

But, the professor might be willing to do it and will certainly serve as the trusted intermediary that might be needed.

You'll have to answer this one yourself. It's similar to asking someone else for a favor.

If you read the rules of your library, chances are they'll say that the person who borrows the book is liable for it if it is damaged. This also applies to ILLs. For example, if you spill water on the book by accident, it's your professor who gets in trouble. If it's just a fine then you can offer to pay for it, but there could also be other consequences (such as the incident being noted on your professor's record, or the other library might be reluctant to lend to your institution's library in the future).

It's also possible the library has explicit rules against a borrower giving the resource to someone else - you will have to check.

As the other answers point out, this depends on your relationship with the professor. But it might be possible to get access to these materials directly either as a "community" member of an academic library (ask) or even through a public library (again, ask).

Since you are an alumni of at least one university, check with those libraries about the services they offer to alumni. Inter-library loans are not usually included in library alumni services, but many libraries are happy to make exceptions for good reasons.

As well as asking about inter-library loan, ask the library if it is available as a e-resource.

It depends on your relationship with the former professor, however, I've had the opposite experience: of late fees incurred when I was a research assistant for a professor who borrowed my ILL book. (The fees were paid by the professor later, so the result was fine.)

Are there any libraries near you that also do ILL? My local public library uses ILL.