Here the circles define the finished dimensions of the brick wall, the point where the roof begins, the height of the ridge and the width of the chimney. The use of geometry appears to determine the design of the new wing, not its framing - a change possibly brought on by contact with the builder's English neighbors.
The circles laid over the south elevation of the original house determine where the roof sits on the walls, the height at which the ridge is located - information needed for construction.
These are interesting pictures. But how would these circles have been actually used for construction and design?
Here are some preliminary thoughts.
The circle can easily be scribed on the ground, for layout as in the foundation.
Wood bents - posts and beams - would have been cut and assembled on the ground, marked -'scribed' - and then taken apart, moved to the site and reassembled.
The original house has a wood frame enclosed by a brick skin. The wing appears to have corner posts. The rest of the wood frame is not clear. Most likely the brick walls cover a wood frame. Perhaps one house-wright/mason framed the house and another, the wing.
Brick walls are laid only once. Handmade brick probably was not always true to size in 1737. The mortar would even out the discrepancies in the coursing if the mason knew how much to apply. If a dimension were constant - as in the distance from one point to the next on a circle array - a pole that length could be used as a template. The distance from the center of the foundation to the edge of the first floor (one length of the hexagon inscribed in the circle) would be easy to establish with a rope, chain, or pole. That pole could then be used much as is a story pole today. If the pole were mislaid, the dimension could easily be re-determined.
Drawings:1934, HABS, Adam Van Alen House, Kinderhook, NY, E. J. Potter, delineator