Edit: If you want full details on this adder, pages 60-63 of the manual  explains how the adder sums the voltages, getting 54V, 104V, 154V, or 204V from the sum. The digit resolver then turns this voltage into a sum bit. (The carry bit is easier, as it can be generated by a threshold.) You can look at the schematics  if you like vacuum tubes; the digit resolver is schematic #200.
Also for anyone who has trouble accessing the page you linked:
They were almost the size of then-contemporary transistors; just a tad larger. They were made in a special process (inside a vacuum chamber). Pretty much the zenith of the technology, and showed what it could be pushed to do.
• at 10 watts it uses several times as much power as my computer.
• what company is this National Union Electric Corporation? Apparently a HVAC company that merged with Eureka in 1959 and is now part of Electrolux: http://www.fundinguniverse.com/company-histories/the-eureka-... — how the mighty have fallen.
• You can hook up the same tube as an oscillator, a flip-flop, an XOR (“inhibitor”) gate, a 3-input AND, or an adder!
• it runs on 300 volts—not a TV flyback voltage but enough to give you a good strong jolt. Electronics has calmed down a lot since the wild 1950s.
• no numbers are given for price or reliability. I assume reliability would improve by reducing the number of tube filaments in your device.
• 300 nanosecond carry delay! That's pretty fast for 1950s logic! I wonder if it was that fast as a flip-flop, too.
Not only an adder, you could probably mix audio channels and create interesting effects.
In the Space Age, building 300VDC power supplies was basically a prerequisite for any teenage electronics hobbyist.
The BG2 tube was probably not mass produced.
This looks like a design that NU made prototyes of and was willing to manufacture if enough interest developed.
You'd probably need feedback to mix audio channels with it, since it wasn't designed to be linear.
The response was nothing short of amazing, 30+ people responded, about half we made deals with an several of those are now regulars. Way beyond anything I imagined.
When you don't need to add, just count and compare, you can use a LFSR, as the Atari video generator infamously did for sprite positions.
Modern computers, even back to the 1970s, usually instead use lookahead carry to get logarithmic carry delays instead of linear ones.
If you use a compound or otherwise non-binary base, carry logic is different, but the same basic options still exist. The 8008, 8080, 8086, i386, and amd64 instruction sets have “auxiliary carry” bits for use with BCD arithmetic; a separate DAA instruction adjusts the binary addition result to be a BCD result. The weirdest thing is that the 8086 added DAS, DAM (?), and AAA.