"...Mr. Babbage believes he can, by his engine, form the product of two numbers, each containing twenty figures, in three minutes." —L. F. Menabrea, Sketch of the Analytical Engine, as translated by Ada Augusta, Countess of Lovelace.
Charles Babbage (1791-1871) never completed his half-baked project, the Analytical Engine. Had it been built, Babbage's Engine would have ushered in the computer revolution one hundred years early. Alas, the machine was either too mechanically complex for the time, or the financial and political will were lacking, and the first digital, programmable, and Turing-complete machine, the Zuse Z3, would not be built until 1941. Nevertheless, the very design of this project was so powerful that it introduced the modern ideas of assembly language, microcode, and the computer programmer. Not so half-baked after all.
Babbage used base-10 logic in his design, but there is no particular reason why he couldn't have built a mechanical computer using base-2 logic (as Zuse later did). I posit that such a machine would have been much easier to build if he had used rod logic and Boolean operations.
"I have been looking at cruder devices analogous to the Babbage engine, but building more directly on the methodologies that have been developed for integrated circuits; i.e., using binary instead of base-10 logic." —Eric Drexler, Rod logic for molecular computing, in Proceedings of the 1989 NanoCon Northwest regional nanotechnology conference.
Drexler's rod logic is based on sliding rods which cross at right angles. Rods having nubs placed at fixed locations along it can inhibit the movement of crossing rods when the nubs on each rod interfere. Diagrams showing rod logic gates can be seen in Ralph Merckle's article, Two types of mechanical reversible logic, in Nanotechnology, vol. 4, 1993.
Starting with simple gates, working up to an ALU, and maybe even a simple programmable computer, would be a fine half-baked project. And, of course, the whole thing would have to be done in brass with flourished lettering inscribed on.