This is an interesting topic. Without quoting anyone else's messages,
but acknowledging that I am drawing on other people's ideas, here are
some suggestions.
Memory. I'd go for mechanical. Either decimal (there were mechanical
calculators around at that date that had decimal and even duodecimal
wheels for registers) or binary. I think the first binary mechanical
memory was built by Konrad Zuse in the 1930s. They've reconstructed it,
along with the rest of the Z1, at the Technical Museum in Berlin. It
could easily have been done using Victorian precision engineering
Logic. I'd still go for relays. I don't know whether there were relays
as such in 1900 - my electrical engineering textbook of 1892 doesn't
mention them - but the coils, cores, and components to make switches and
actuators were all available.
On the other hand, there's a lovely _programmable_ mechanical
calculator, from about 2 years before Zuse, preserved at the IBM museum
in Sindelfingen. This was just like any mechanical calculator of the
day (decimal wheel registers), but with several accumulators. You could
program nine steps of something resembling horizontal microcode on a
plugboard somewhere; it then sucked in hundreds of punched cards and
processed each according to the nine steps. There were conditionals in
there - "if the card has a hole in position foo then add the total from
accumulator 1 into accumulator 3" was I believe a possible instruction.
This machine was made by Dehomag - the German Hollerith Machine Company
- who were later bought out by IBM. Very fitting, since this machine
was a typical IBM box - horrible processor but data throughput to die for...
And despite the 1934 date, I expect it could have been built in a 1900
precision machine shop.
Philip.