Thinking about a board game based on this machine. Just jotting down a few ideas:

Move proceeds around a board as player assemble pieces to construct their machine. The first to make it to the patent office with a working analytical engine wins. Players first make a difference engine and either patent it or build one by commission. Pieces from difference engine funds analytical engine.

If a player builds one for patent and not commission, he or she can use all the pieces of the difference machine to build analytical machine. Everyone else can only reuse their original (before being commissioned) pieces.

There are three pieces - spacer, cog, and rack.

The rack is a piece that allows the machine to be built 'up' a level. Presumably, the more racks, the more variables that can be calculated at once, but also the longer it takes to finish the machine.

The spacer allows the machine to be built 'out' a level. Presumably, the more spacers, the more decimal places that can be calculated for each variable. Once a spacer is added, it is added to every level.

Cog is a bit of machinery that fits on the grid created by racks and spacers. For example, a machine with 5 racks and 8 spacers would require 40 cogs to complete.

There would need to be a funding source, so I guess each player is a hobbyist until if and when a company commissions a machine to be built. Each would have different specifications and different advantages (more cogs per turn, alternate build rules, etc.)

Upon completion, they would get paid and then start to build the analytical engine.

Need to add in bonus or random elements like:

Ada helps your project (build at twice the rate)
Building trouble (destroys rack or spacer)
Lose funding. Keep your pieces, lose those purchased by company commissioning the machine.
Cogs go bad and must be replaced. or only red cogs go bad.
An entire rack goes bad and must be replaced.
A green rack goes bad or a red rack goes bad.

Piece would be two different colors to distinguish pieces built by you, the inventor, and those purchased by company commissioning you. Say Green are built by you and red built by company. When difference machine is finished, you can use only the green pieces to start analytical engine. (Thus if you built it entirely yourself, you could reuse all the pieces.)

Analytical Engine would need fourth piece, don't know what to name it. Maybe replace cogs or modify cogs. Maybe analytical cogs cost more, but difference cogs can be modified very faster. For example in one turn you may get an analytical cog or convert three difference cogs to an analytical cog.

Building rules:

Rack pieces can only be removed from the top down, not the bottom up.
Removing a spacer affects every rack. The cogs lost cannot be reused. (Someone may remove a spacer in order to finish their machine faster.)
When you start on your analytical engine, you must add at least one rack and one spacer.
A difference machine must be at least 6 racks by 6 spacers.

Unless directed by card or company paying you (difference engine only), you can only add the piece you land on while moving your token around the board. If you are being paid to build, it must be a red piece. If you are going it alone, it must be a green piece.