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It could be some kind of AM radio antenna for crystal sets... I remember reading something once about wrapping speaker wire around a paper towel tube to make an antenna. At least, all the terminals on the side give me that impression.
And an oscilliscope from the 50s is pretty old, as far as scopes go. The oldest one I've ever seen is our HP from the 70s.
However, transformer is also an interesting idea. It looks like you could vary the output power by rotating the inner winding. With really large voltages, like 120v, 240v or more, it might be useful for running some small experiment or something.