The Calculus of Finite Differences by L. M. Milne-Thompson is a classic. It covers a great deal of elegant and useful [1] mathematics that isn’t widely known, at least not any more. For a taste of the subject matter of the book, see this post.
The book is now in the public domain—at least in the United States—and so you can find scans of it online. The copy I downloaded and a hard copy that I purchased both have some problems. I’m writing up how to fix the problems in case anyone else runs into the same difficulty.
I bought a hard copy of the book because I prefer reading from paper, though I also like having PDFs for reference. I thought I would patch the PDF scan by scanning the pages of my hard copy corresponding to the pages missing in the PDF. However, the scan and the paperback version have the exact same errors.
Pages 1, 2, 31, and 32 are missing. Actually, pages 1 and 2 are included, but not where you’d expect. After page 396 comes page 2 and then page 1. Pages 31 and 32 are completely missing.
I went back to the Internet Archive and found a different scan of the same book, one that appears to be complete.
My guess is that the errors went back to the first printing in 1933, and were corrected in subsequent editions. Other than correcting the missing pages, I don’t know of any changes between editions.
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[1] The author describes his motivation for writing the book as follows.
Two distinct reasons impelled me to undertake this work. First, in my lectures at Greenwich to junior members of the Royal Corps of Naval Constructors I have occasion to treat certain aspects of difference equations; secondly, the calculation of tables of elliptic functions and integrals, on which I have been recently engaged, give rise to several interesting practical difficulties which had to be overcome.
By coincidence Mathologer recently posted a short visual introduction to the subject: https://youtu.be/4AuV93LOPcE
I’ve been raking my brain since that video came out. I cannot remember for certain whether we touched on this in college (Engineering, late ‘80s, Latin-America)… but it sounds incredibly familiar. I suspect we skimmed it for a couple of weeks. At what level is this commonly taught today is n the US?
Based on my experience, I’d say it’s likely that a student in the US would get a glimpse of interpolation. And they might see a finite difference equation coming out of a power series solution to an ODE. But I don’t think it’s likely that a student would see the calculus of finite differences presented as a coherent subject.