Not fair, nancita. I clicked on your post expecting photos of a "Cool Stairway". Hmmmm. Did you notice the coolness before you removed the carpet? Is it a draft? What is below the stairs?

Before thinking insulation, do some checking. These old girls were usually sealed, but not insulated. The layer of air trapped between the interior and exterior wall was the insulation and over time, the seal of the tar paper, etc has broken down and you get drafts. Insulation is fine, but you have to make sure there is NO knob and tube wiring present in the cavity. K&T wiring was designed to "breathe" and insulating around it can cause overheating and possibly lead to fire. My 1885 was MISERABLE for the first two winters. You could see your breath everywhere until I got all of the drafts sealed up this year and it has been much warmer.

You, know, I actually thought it was a terrible title. Will change it. Thank you.
We did notice the draft even when the carpet was on the stairs. Interesting about the ktw. Yesterday, the energy audit guy said the first floor had fiberglass insulation. It was clear the previous owners remodeled that floor. The second floor has carpeting and paneling everywhere. The energy guy drilled a couple of holes in the upstaris walls and said the drill went easily into the wall, then there was a space, then another inner "wall" of who knows what, possibly as you said, tar paper?
The townhouse we are renting until we redo our home has the same problem. The temp gets significantly colder when approaching the stairs.

Heat rises, and cold air falls, right? Stairwells are corridors for air movement. If the upstairs is not insulated and is cold (as ours is, not even heated) then any heat that gets into the stairwell just goes up and disappears. Cold air, I would hazard a guess, also falls down the stairwell. If the stairwell ends on your main floor, it would discharge onto the main floor. If it can go further down instead, it will,

Our stairwell has an exterior wall, beadboard badly insulated if at all. Stairs often do go up outside walls. That can be part of the problem. But the main problem is their connection to the floors that may not be heated. Doors top and bottom, to the extent this is possible, would help; but of course it may not be. Regardless, I suspect the solution lies in looking at the overall circulation of air in the house, not just the insulation of the stairway.

Karin L