The Geometry of Fences, c. 1830

Asher Benjamin's The Architect or Practical House Carpenter, published in 1830, includes a plate with 3 designs for fashionable fencing,  2 for gates. The lower drawing also includes a post.



While Benjamin includes a scale between the middle and the lower illustrations, he gives no other dimensions or information. He assumes the reader will know how to lay out the design.

The 2 right hand designs are repeated diagonals. quite simple to draw: cross your rectangle, lay in the horizontal and vertical center lines, embellish as you wish.
The bottom drawing is the fence for the gate above and shows the post and its ball.

What about the fence with curved balusters and the gate below it with rectangles and crosses?




First: the fence with curved balusters: 
The center of the rail is the center of the arc. The extended arc becomes a semi-circle  whose radius appears to be the distance to the edge of the bottom rail from the center and the height to the post below the ball and its base.

The arc determines the curve of the baluster. The circle using the same radius, centered on the baluster, follows the reverse curve. Both balusters are shown in red. 
Using Benjamin's scale the balusters could be cut from boards about 3 feet long and 8 inches wide. They are all the same shape.




The circles intersect  - see the vertical dashed lines in black. That intersection gives the spacing for the balusters.
Drawing the next circle using the point where the first circle's circumference crosses the center line, adding the vertical at the intersection, the placement for the balusters, noted in red. continues.
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The gate seems at first glance to be more complex.

 
In my exploration I decided the size and the structure of the gate was known - its width, the framing members, the depth of the bottom rail. The black rectangle outline is what's left - the space for the fret work.
The diagonals are easy to draw - corner to corner. The space can be easily divided in half horizontally and vertically. I have not noted those lines.
But what about the inner rectangle?  The little corner squares?







Two radii are drawn; both A and B are arcs the height of the space, their centers are in the bottom corners, they arc from the top to the bottom - follow A and B's arrows.  They cross at C which is both the vertical center of the rectangle and the depth of the cross brace.







Reverse the arcs.  Now A and B land on the bottom rail  and mark the placement of the vertical braces. C marks the horizontal brace .







The horizontal and vertical braces are noted in red.

The rectangle of the space could be larger or smaller without changing the way the design developed or the general appearance. The proportions would still relate to each other.


The reader in 1830 - probably a house carpenter - would not have required my explanation to copy or adapt the  designs for his own use. He would have learned Practical Geometry as an apprentice. He would have read the design development automatically; he needed no words of explanation.
He might have drawn  his own small diagram on a board. Then he would have drawn the arcs full size on his framing floor - or table, as this is not very big - and found his lengths. The diagram would have remained until all the parts were made.


Plate XXXIII, Asher Benjamin's The Architect or Practical House Carpenter, 1830, L. Coffin, Boston. From the Dover Reprint first published in 1988.


This Asher Benjamin pattern book especially interests me because a diary was written by a local farmer in this period. Its author notes that his friend, a carpenter, traveled twice to Albany to buy his copy of the pattern book when it was first published. The details in several local houses seem to indicate that the joiner worked directly from Benjamin's plates.