We define programmable machines as those that can do:
ie the Turing Model of Computing.
Here we see the UCLA's 1948 Mechanical Computer performing a calculation.
To me this appears to be a Differential Analyser.
My question is: Was UCLA's 1948 Mechanical Computer programmable, or merely a calculator?
From what I know, and what the video supports, this is 'simply' an analogue computer. Analogue computers are not programmable in a sense like we use the term today. They are more like Construction Sets with predefined blocks and mounting plates. In this case like a highest quality Meccano-like set. Analogue computers work, as their name say, by building a device working analogous to the problem in question. When build/configured, they represent a fixed function machine. Depending on the setup with fixed values, or parameterized input.
Using a 'premade' analogue computer is not different than building the same machine from scratch. It just simplifies the construction process and enables repetition due its premade parts, designed to ease construction (*1). In this it doesn't matter if made from mechanical parts, to be bolted into a frame (like the example) and driven by some motor, or a structured set of amplifiers and resistors in an electronic analogue computer to be wired up according to the problem (*2).
In either way, the term programming, as it used today with computers, is at best borderline, but usually not applicable here. They are fixed function setups.
*1 - Which is the very definition of a construction set, isn't it?
*2 - Note that this is to be distinct from early digital, patch (board) driven machines like D11 or ENIAC that had to be wired up as well.
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