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I have not been able to find a librarian who remembers the old system. So the descriptions here are purely from my own childhood observations, although they were probably just as valid (or otherwise) for some years before and afterwards. If you can supply further information or an illustrative photo, please get in touch!
In the middle of the 20th century, when I was growing up, libraries seemed to organise their loan system extremely efficiently, even though there were no computers.
Understanding how they did this, first requires descriptions of readers' 'tickets' and the inside front covers of the library books.
In order to join the local public library, we had to show some form of identification to confirm that we were in a household which was contributing to the library by paying rates - the precursor of council tax. We were then issued with library tickets. (Yes, tickets, not just one card.) We then became 'members of' or 'readers at' the library.
My recollection is that every adult member had two tickets for fiction books and two for non-fiction (factual) books, but this may have varied from one council to another. My family was in Edgware, north London, then Middlesex.
The two types of ticket were of different colours and children's tickets were of yet another colour. The tickets were made of folded card, stuck together at the bottom to form corner pockets. They showed the reader's name and address; whether the ticket was for fiction, non-fiction or children's books; and the name of the library.
Library books were instantly recognisable from their inside front covers.
Glued to the bottom left of the inside cover was a pocket very similar in style to a reader's ticket, but giving details about the book to which it was attached. Inside the pocket was a detachable card giving the same details. On the right was a glued sheet giving details of the library and showing the 'due back by' date stamps from previous loans.
There were card indexes for books, by author and by general topic. In practice, though, choosing a library book was normally a matter of just browsing the shelves. Books could be reserved at a cost which included the cost of postage, as that was the only way to let people know that their reserved book was in.
Once we had found a book we wanted, we took it to the librarian at the desk. She - yes, it always seemed to be a 'she' in those days - first took our readers ticket and then opened the front cover of the book. She took out the loose card in the fixed pocket and slotted it into the readers card which she kept. Then she stamped the date sheet to show the date when the book was due back. This was normally a fortnight ahead. We then took our borrowed book away with us.
The librarian filed the readers card (of information about the borrower) containing the book card (of information about the book) under the date that the book was due back. The filing cabinet must have been a perfectly ordinary card filing cabinet, but all I ever noticed was the current drawer which had been taken out of the filing cabinet and placed on her desk. There were dividers in the drawer separating out the dates.
A fine of 1 penny per day was imposed by public libraries for 'late returns' of books.
When the fortnight ahead came round, if the book had not been returned, some sort of note was slotted into the readers card to notify that a fine was due. I have a vague recollection that reminders and fine notifications were sent out by post after six weeks, but I am not certain.
To return a book, the procedure was reversed. The librarian noted the due date from inside the front cover and so located the readers card in the drawer of the filing cabinet on her desk. She removed information about the book, slotted it back into the pocket of the book and gave back our readers ticket. With this, we were able to borrow another book.