You should be able to see that this procedure is ideal for use by a machine. It is also incredibly well suited to a mechanical implementation. All you have to do is build an adding mechanism and repeat it for each of the difference sums that you need. At the start of the calculation you would set the number wheels the values the differences and then simply turn the handle to get the next result and the next set of differences. An entire table could be produced by simply repeatedly turning the handle.

This is the sort of machine that Babbage designed and named a "difference engine". He built a small demonstration model, Difference Engine 0, in 1822 and on the strength of this was awarded the money to design and build Difference Engine No 1.

This he attempted to do with the help of Joseph Clement, a skilled toolmaker and draughtsman, but to be successful he had to push the precision of the metal working of the time to its limits. Babbage was a good mechanical engineer in his own right. He used a lathe, built many test pieces on his own and studied the manufacturing methods then available to see if any could be used to mass produce the accurate parts he needed.

He invented a mechanical design language for the trains of gears and linkages that his machines needed and this enabled him to refine his designs to eliminate redundancies and reduce moving parts. This isn't far removed from the software and hardware engineering principles used today.

You should refrain from thinking that mechanical design of an adder is trivial. To make something that works reliably and in which friction and backlash (looseness) in the gears doesn't destroy the accuracy is very difficult. Babbage's mechanical designs were elegant in the extreme. To give you some idea of how advanced they were he even included a printing mechanism which would produce printing plates by making impressions in papier-mâché sheets - the first GUI interface?  Difference Engine No 1 would have been 8 feet high by 7 long and 3 deep and would have weighed tons.

Difference engine built from original parts by Babbage's son

It was never completed for reasons that seem almost ridiculous from today's perspective. An argument over compensation to Joseph Clement due for moving his workshop to be near to Babbage stopped the project in 1833. In reality it was probably that ten years was too long to sustain such a project without results and Babbage was already thinking other thoughts. A portion (about 1/7th) of the Difference Engine No 1 was put together in 1833 and this at least demonstrated the feasibility of completing the whole machine.

The difference engine was half a step towards a computer because it implemented a simple algorithm rather than just providing the four functions of arithmetic that a human could then use within a calculation. The difference engine was, in our terms, a hardwired computer. Babbage made the leap of the imagination necessary to see that this could be generalised to a programmable device.

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