The Scratch project below is a prototype simulation of the Difference Engine - a fun introduction, aimed at younger learners to educate them about its function.
The engine itself was actually only capable of addition and subtraction, as mentioned here. But I decided to expand my simulation to be capable of all four basic mathematical operations.
I can't pretend to understand the Method Of Finite Differences, but from what I have researched - I gather that if you know the digits at certain places in a polynomial, you can calculate the full answer. (This may be incorrect, I'm not certain. **It's A Level Maths; I've tried.)
However, I do know for certain that the way the engine functions is through the Method Of Finite Differences, and there is a helpful youtube video below, explaining how to put the method to use.