How did the builders of the Parson Barnard House actually use Practical Geometry?
Did they have to draw arcs with twine every time they wanted to measure something?
Probably not. I think that they knew the geometry so well that they could take short cuts in laying out a house .*
Consider a carpenter's education in 1715, when this house was built. A boy would have been apprenticed to a carpenter when he was about 11 years old. Along with woodworking skills he would have learned the fundamentals of geometry: how to use a compass and line for measuring lengths, how to lay out basic shapes with a compass and a straight edge.
After 7 years of training the apprentice became a journeymen. He would have traveled and worked for other carpenters. He would have broaden his understanding of practical geometry and understood how geometry works. He would have been able to skip steps.
The previous posts on the Parson Barnard House** explored the geometry used to lay out the house frame and window locations on the front facade.
The width chosen for the sill, about 18 feet, was the dimension used for the the layout. The room sizes and the overall width and length of the house come from that first length.
The sill length was also the beam length for all the bents.
The diagonals are the Lines which true the rectangle's corners and mark the location of the 2nd fl. beam. See earlier posts**
The chimney mass is the only part of the house with unrelated dimensions. Built of brick, it used the 3/4/5 rectangle.**
The carpenters would have staked a line at one end of the house for the exterior of the foundation 1-2.
Then they laid out the length, 18ft, and staked it 1-2. They laid out a right angle (here shown as a 3/4/5 triangle), and extended that line 1-3 18 ft. - marking the second side of the square; they didn't need to swing the arc. from 2 to locate 3, marking 3 corners of the square: 1, 2, and 3.
The 18 ft. Lines arced from the corners 2 and 3 would cross, locating the 4th point. The carpenters didn't need to draw the arcs, just where they crossed.
Then they would check to be sure it was true, just as builders do today.
Matching diagonals across the square would confirm the carpenters' accuracy. 1-2 = 3-4
The Parson Barnard House geometry uses the length A for its floor plan and its bents. That length comes from the arcs of the length of the bent. It could have been found by 1) folding the Line (about 18 ft long, the length of the bent) in half, and then 2) marking where the arcs cross (dot). A length from the base to the arcs' intersection, the dashed line A, could be transferred to the side of the square, giving the width of a room, the height of the bent. (NB: Geometry requires 2 points in order to draw a straight Line.)
The carpenters only needed to find this length once. They could have marked it on a plank for reference until the frame was complete, then used the plank as sheathing for the walls or roof.
Some of these layouts have been found and saved: Eastfield Village in East Nassau, NY, (https://www.historiceastfield.org/) has some which they noticed on the roof sheathing when the Inn was dismantled and moved to the site. Unfortunately, because we have lost most of our understanding of practical geometry, those doing restoration rarely look for such notations left by earlier craftsmen.
*NB: When I teach, the students and I swing the twine in arcs to mark the corners of a house foundation. It was easy, fun, and exciting as we see the shape come into being.
** see: https://www.jgrarchitect.com/2023/11/the-practical-geometry-of-parson.html
*** Brick walls are built row by row. 2 Lines and the 3/4/5 keep them true vertically and horizontally.