The Carpenter Square and the Compass - The Evolution of Practical Geometry

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On May 31, 2025, I will present Practical Geometry and Carpenter Squares at the Early American Trades Association (EAIA)* conference in Rochester, New York. I will talk about how the use of a carpenter square for many layout decisions  - using a 3/4/5 rectangle instead of a circle - changed the visual character of our architecture.  

 

The squares shown here were made in Vermont c. 1830-50. They now live at the Bennington Museum, Bennington Vermont, and can be seen by appointment.

 

What happened after 1820 when the carpenter square became a reliable drafting tool? When the compass, line, and scribe were joined by an L shaped piece of steel with a dependable, true 90* corner?

 

 

 

This image shows the hand stamped numbers on the earliest squares as well as carefully drawn scales.

 

 

 

 

The square made design and layout accurate with fewer steps. That meant it took less time to draw and measure work. However, little by little the accessibility of a true 3/4/5 triangle became a design element as well. Here's an example.

Robert Shaw's The Modern Architect was published in Boston in 1854.** This is Plate 4, a 'Grecian  Frontispiece'. The main door is laid out as a square.

I've added the scales on the bottom and right side. It is 8-1/2 units wide and tall. The door is 3 units; the pilasters are a half unit. The sidelights are 1 unit; the columns are 1-1/4. The columns' capitals are a half unit.

The red quarter circle arcs of the width cross at the top of the door frame, just below the transom. This layout choice can been seen much construction pre-1820.

 

 

The whole frontispiece is a square plus its diagonal tall. The height is equal to the diagonal of its width. 

This proportion is also used for the sidelight glass panes (see the image above), but not those in the transom.

 

 

 

 Shaw writes that the door's height should be "...over twice the breadth of its height as three and seven feet."* It is also a shape drawn easily by the carpenter square.

 

 

 

 

The door and its columns taken together become a simple 3/4/5 rectangle.    

 

 

I think we architects, crafts people, and historians are seeing geometry when we note from visual observation that a particular building is 'Greek Revival', not 'Late Georgian' or 'Federal'. We haven't recognize how practical geometry evolved during the Industrial Revolution.

 

* EAIA, Early American Trades Association https://www.eaia.us/

https://www.eaia.us/2025-rochester-ny

** Robert Shaw, The Modern Architect, Boston, 1854, originally published by Dayton and Wentworth, republished  (unabridged) by Dover Publications in 1996.

*** Shaw, The Modern Architect, page 63.