Doorbell circuits are separate and different from house wiring. Therefore it is not 'knob-and-tube'. It's just old.

So when you removed the old doorbell, did you remove the transformer as well? They are usually not in the same place.

Knob and tube isn't "cloth covered". As weedy points out the door bell wiring isn't ever going to be knob and tube.

Most likely the transformer is elsewhere. What was removed from the wall was the chime box. If you only have a front door bell (which appears to be the case because you have only two wires there), you can connect up just about any door bell chime here (provided none of the rest of the circuit is damaged). Door bell circuits are isolated AC to it matters not which wire is which.

Here's the old bell I disconnected. Is that the transformer in he middle? The new bell has a transformer too.

If that's not the transformer can I just skip the transformer part and leave the original one and just hook up the new chime box? It's just a front door bell. There is an additional chime upstairs for the same door that we haven't touched. But as I've turned off the circuit it isn't currently working.

No, that's not the transformer. The transformer is probably located in the basement or attic.

Just hook up the two wires to the new door chime. It doesn't matter which wire is which.

Yep, that thing in the middle is the solenoids for the striker. This unit was designed for two door bell buttons (though you can wire it for one). The front door is wired to fire one plunger which hits one chime and then bounces with the spring to the other to make the "Ding Dong" sound. The other plunger just goes "dong" (or ding I guess) for the back door.

Get your new chime and follow the instructions for hooking up only one (front) door. As we've stated, it doesn't matter which wire is which but you need to connect it to the right two out of three screws on the doorbell if it has a two door feature.

Thanks so much! We connected the new bell and it works perfectly!

W or wires how do we attach the new one when we have four wires

Presumably you didn't take photos or notes when you removed the old doorbell. I strongly suggest you do so next time.

Most likely only 3 of those 4 wires are required. In most cases, you'll have a wire for the front door button, a wire for the back door button, and a common return to the transformer.

Typically there will be 3 terminals on a doorbell. (I'm assuming here that this is a traditional mechanical doorbell.) One will be marked F or Front, one R or Rear, and the third something else -- usually T or Transformer. (It might also say C or Common.) You may have to refer to the doorbell installation instructions for these designations.

Have someone hold the front door button while you experiment with touching various pairs of wires to the T and F terminals. When you find the pair that ring the bell (ding when you touch, dong when you let go), connect those wires to those terminals.

Now have your assistant hold the back door button, and touch each of the remaining two wires to the R terminal. If you get a ding, connect that wire to the R terminal.

If you don't get a solid ding with either wire, or if you just get a sort of clicking or clunking or humming noise, reverse the first two wires (that is, connect the one now connected to F to T, and vice versa). Then try the above test again.

Put a small wirenut on the unused wire, or insulate it with plastic electrical tape.

If this doesn't work, post back, and we'll try to help further.

When we attach the Rear door wire, the doorbell constantly rings (we don't have a backdoor button, When we leave it off and just have the F and T wires attached, the front doorbell doesn't work. Should we attach the R wire to the F terminal too? Everything used to work until recently.

HU- It’s always best to start a new thread rather than piggybacking on an old one. You’ll get better answers if people don’t have to work through all the prior posts that only marginally relate to your situation.