The only thing that I could add to your fine explanation is that your comment of the slope being 9\11.
If you were using the square to layout the cuts bottom and top of the rafter you would use 9.81 (plus a little bit more) inches in 12 to arrive at the 108 inches rise in 11 feet of run.
I hope the quote comes out well.
Anyhow, I think the simplest or least derived slope ratio is the way to go when at work. In this case, 9 ft. rise over 11 ft run. Hold 9 in. over 11 in. on the framing square and step twelve times for rafter cut length. Take a minute and examine the outside back face scales of the traditional framing square, inches divided by twelths. In a time when roof pitch or total gain was specified the simplest modeling and calculation can be accomplished on the square directly. Lets say a building is 28-4 wide with the roof rise of 11-5. Using the back face of the square hold 14 2/12 and 11 5/12 and step twelve times for rafter layout. For layout with the best accuracy use the square for setting trammels or large dividers. Precise length or slope in base 12 is not necessary.
On the cost I bid $1,235 with no factual underpinning. Old builders manuals such as Trautwine and Audels have costs at their time of publication, it would be interesting to find a close contemperary to 1875.