yet another gear loop

 It turns out that there is another gear loop in the Plan 27 design that needs to carefully designed to mesh correctly.

The long pinion is connected to the anticipating carriage (which does addition and subtraction) by an intermediate transfer pinion. But sometimes during complex operations like division, an additional reversing gear needs to be inserted into the chain. Babbage does that by vertical motion of the transfer and reversing pinions, with the reversing pinion being double height. Here are the two configurations:

non-reversing

reversing

Although clearly all three gears are never meshed as shown  in the plan view, the teeth need to be aligned in order for the vertical movements to happen without tooth conflicts. 

There are far fewer degrees of freedom for axis positions here than for the 5-gear mesh described in the earlier posting. Given the constraints on locations of the gears to avoid interference, the first attempt failed to produce any solutions. 

But there is another degree of freedom: we can vary the number of teeth on the transfer pinion and the reversing pinion, as long as they all have the same tooth ("diametral") pitch as the long pinion and the carriage. For our prototype, the DP is 5, or 5 teeth per inch of gear diameter.

To search for solutions I modified my stochastic gear analysis program to be able to vary the number of teeth (within a limited range) as well as the locations of the gear centers.  It produced only a few solutions, and thankfully at least one for each of the two carriages avoided interference with any other gears. (The source code is on GitHub at https://github.com/LenShustek/AnalyticalEngine in the simulations/gear_meshing director.)

Here is the final proposed layout, based on a simplification of Plan 27, for a prototype which can handle multiplication and division and has a total of 12 numbers in the Store:

There are several other changes compared to the version in the previous post:

• The Mill pinions to the rack now connect to the Px1 pinions instead of the movable long pinions. That takes one gear out of the chain in reading and writing to the Store. It also means that the Mill digit wheels and the Store digit wheels turn in opposite directions, so the numbers don't have to be reversed to make transfers work right.
• A single Signwheel is proposed to handle all the signs in the Mill, together with some simple state information based on the microcode ("barrel") execution. That means that the Mill digit wheels and carriages don't need sign wheels on the top of their stacks. (In they Store they do, of course, because they record the sign for the sign-magnitude representation used there.)
• Two linked 20-position counter wheels are proposed:
• One is used a simple loop counter during division.
• The second is used to set the number of implied decimal digits to the right of the decimal point, which affects final processing for multiplication and division. It is "given off" to the other wheel as that processing happens, and then restored in a subsequent cycle for use in later instructions. Babbage expected that it would be set by the machine operator at the start of a session, and we do the same.