First generation was just to keypunch the data I think. The system sorted the charges by call number and made it available on green-white printouts that arrived the next day. The system also generated overdue notices-- or at least the second generation did.
Second was the big blue Mohawk Data System terminals, with punchcards for each book and holepunches in the Columbia ID cards; these were in use when I arrived in summer 1970, but not for long because we were "carding" books that summer. They fed data to a controller and a set of big tape drives located in 406, the Burgess office. The tapes were dismounted and taken to Computer Center at closing time. The printouts were eventually supplemented by microfiche which was a lot smaller but took an extra 12 to 24 hours to arrive, compounding "the lag" that made us crazy in trying to find books.
During this period Paul Peters rewrote the code from whatever it was (assembler?) to PL/I. I think John Leide was the person who wrote the original code. Paul said it was written to run fast but was very hard for humans to read, 'spaghetti code' typical of that time period. I recall we could change the text for notices to users but only with exactly the same number of characters, like patching a binary now, as if it would have been too big a job to make it print a different number of bytes.
Third generation was much smaller black terminals that could read barcodes and holepunches. I have one of them in my office and it says July 1983 on it. These also wrote to tapes in 406 using the same cables. Sometime around 1986 I learned Wylbur and we were able to get access to the data files via terminals, cutting "the lag" waiting for printouts and fiche. Data was still updated only in one batch run per night.
I could go on...
I'm hoping to get a more complete story / timeline of this very large aspect of computing at Columbia. Circulation is just a piece of the pie. Another major project was conversion of the huge 200-year-old card catalog to online (Columbia has one of the largest research collections in the world). This was a two-pronged effort. First was the conversion of Acquisitions from cards to computers; second was retrospective conversion of existing catalogs, which started about 1978 (?) and continues to this day. I presume, but don't know, that computers were involved in the massive Dewey-decimal to Library-of-Congress conversion of the mid-1960s -- if not, then at least this was a major factor in the later automation of Acquisitions. Other aspects of library automation include everything you see at the Columbia libraries website: databases, search engines, online books and journals, images, interlibrary services, and assistance in many forms.
Although this has absolutely nothing to do with computing, I thought I might insert a favorite story from my student library assistant days here. Every so often the Library declares an amnesty on overdue books, when fines are forgiven and no questions asked. During such an amnesty in the mid-1960s, a very VERY old gentlemen sheepishly approached the Burgess-Carpenter circulation desk and handed in a book he had checked out as a student... in 1888.