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

 

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 expect I will be introducing Practical Geometry and then explore how the use of a carpenter square began to change the visual character of our architecture. I hope to see tool collections and hear other members' thoughts.

 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?

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

 

 

 

 

Here you can see the hand stamped numbers on the earliest squares as well as carefully drawn scales. Were the scales as important to the builder as the true 90*angle?

?

    

The square made design and layout accurate in fewer steps. Units (inches and feet) were uniform, corners were square, always 90*.  A job could be drawn, measured, and laid out more quickly and accurately. However, loosing those steps also changed the proportions. I have written about how this can see seen in vernacular housing design.**** I wanted to learn how an architect might have used the carpenter square. Robert Shaw was a good choice  because he wrote a book.

 

Robert Shaw's The Modern Architect was published in Boston in 1854.** 

 

 The pattern book's frontispiece shows the tools of the builder and the architect. The original drawing is an engraving which is quite dark. The color was added when the book was republished in 1995.

 

In the foreground is a large compass, probably used for stepping off. The architect holds a little one. The architect and builders are shown conferring, syncing the construction dimensions with the drawings .

 

 

Here is Plate 4, a 'Grecian  Frontispiece'

Where did Shaw begin his design?  Conceptually the design surrounds the door, giving it emphasis. So I began there.

Shaw himself stated that the door's height should be "...over twice the breadth of its height as three and seven feet."*** 

I have added the scale below the door: 3 units for the door's width. Then a half unit for the columns on each side and a full  unit for the width of the sidelights. 

These proportions follow those recommended by James Gibbs in 1732. ****

 

 

 

Was Shaw using 'circle geometry' for his layout? I don't think so. The circles don't offer much information. 

While the layout is 2 circles tall, the 12 points around the circumference of the circles give only the height, the width of the entry including the side lights, maybe the location of the transom. Note the arrows.

 

 

I think Shaw used a  simple geometric pattern that is derived from the circle, but which doesn't need to start with a length - a radius - and compass. It starts with the square which is easily laid out by the carpenter square. 

The width of the door and its sidelights was the dimension for a square. That shape was easy to lay out and make true with a carpenter square. Beginning with a length, he set up the corners with the square, added the lines for the 4 sides,  trued the box with diagonals. The diagonals used to find the additional height comes directly from the square. Done.        Note the arrows.  

Was there a name for it? Not one I've found.  It's basically a 'square and diagonal geometry'.

 

 The door, its transom, sidelights, and columns are also a square.

Here the quarter circle arcs, based on the width, cross at the top of the door frame, just below the transom. This layout, creating a slightly smaller rectangle within the square, was often used in layout and design. ****  I think here it is incidental.

I've extended the scale across the bottom and up the right side. It confirms the geometry.

The whole frontispiece is 8-1/2 units wide and 10-3/4 units tall. The door, the pilasters and the sidelights are 6 units wide; the columns are 1-1/4 units each. The columns' capitals are a half unit tall. The entablature is 2 units; the pediment, 3/4 of a unit tall.

Each unit and its parts could be stepped off with a compass. In 1854 the length could also have been stepped off in 12 inches intervals as marked on the carpenter square. As shown in Shaw's frontispiece in his book, it seems the builders used both.

 

The geometry used for the door and its parts is also used for the overall size: the height of the frontispiece is equal to the diagonal of the square.

The lightly drawn dashed line is the arc of the width of the door, showing how it lays out the square. This geometric proportion is also used for the sidelight glass panes (see the image above), but not those in the transom.

 

 

When we architects, restoration trades people, and historians note from visual observation that a particular building is Greek Revival, not Late Georgian, we are seeing geometry. I think we are recognizing, even if subconsciously,  that the rhythms, the proportions of  Federal architecture are different from the Greek Revival proportions shown here.   

 

* 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. 

****    For more information about James Gibbs' use of the door width as a unit of measure see: https://www.jgrarchitect.com/2025/01/james-gibbs-and-rockingham-meeting-house.html

            For more information about the square and its rectangle see:                                          https://www.jgrarchitect.com/2023/11/the-practical-geometry-of-parson_20.html

           For information about buildings using the 3/4/5 triangle for layout:

  https://www.jgrarchitect.com/2014/03/railroad-warehouse-frame-c-1850.html  

https://www.jgrarchitect.com/2014/10/the-cobblers-house-c-1840.html  

https://www.jgrarchitect.com/2013/10/1820s-farmhouse-north-of-boston.html 

  

Practical Geometry Lessons, Lesson 5: Rectangles

Today these skills are not required knowledge for builders. We have steel carpenter squares that have true 90* corners, as well as levels and  lasers. 


The carpenter squares shown here are some of the earliest made in the States. They were made in Shaftsbury and N. Bennington, VT, 1825-60. Some are on display at the Bennington Museum; all are available for study.

The 1503 woodcut at the end of this post includes a square being used for a layout. That square might not have matched the square of another builder.

Practical geometry taught how to 'prove' that an angle was 'true'. Carpenters today still make their work 'true'.  

This "Geometric Problem'  and its solution was particularly important when carpenter squares were not necessarily true: the square corner was not always accurate, not dependably 90*.

This is the end of Lesson 5.
The carpenter's assistant who masters these problems is now ready to assist in layout and framing. Maybe he (no recorded 'shes' that I know of) will go on to learn design. 


For more ways to draw a square  see Drawing a Square, Parts 1 and 2.
Part 1: https://www.jgrarchitect.com/2019/12/practical-geometry-drawing-square-with.html
Part 2: https://www.jgrarchitect.com/2020/01/practical-geometry-drawing-square-with.html



 
After-thoughts and questions:

How did carpenters carry a 10' rod?  Did the rod fold, or come in pieces that could be connected with pegs,  perhaps leather sleeves?
I have seen one in a medieval print, part of which is shown here.
The builder on the left uses a square for layout. Below him is an axe, a saw, and a level with a triangular hole and a plumb bob.
The builder in the center holds a 10' rod.   The landscape to his right is the site where he will layout a building, or perhaps land.
 A 'rod' when used in land surveying is 16.5 feet long. It is also called a 'perch' or 'pole'. How did they carry that awkward length? 'Links' and 'chains' are also used in surveying, so perhaps a rod could be a length of chain.

 Woodcut by Gregor Riesch, Margarita Philosophical, published in Baden,1503.

 

The posts in this series  Lessons 1-7  are :

 https://www.jgrarchitect.com/2020/04/lessons.html

 https://www.jgrarchitect.com/2020/04/practical-geometry-lessons-2.html

 https://www.jgrarchitect.com/2020/04/practical-geometry-lesson-3.html

 https://www.jgrarchitect.com/2020/04/practical-geometry-lesson-4.html

https://www.jgrarchitect.com/2020/04/practical-geometry-lesson-4b-old-first.html

https://www.jgrarchitect.com/2020/06/practical-geometry-lessons-lesson-5.html

https://www.jgrarchitect.com/2020/06/practical-geometry-lesson-5-addendum.html

https://www.jgrarchitect.com/2020/08/lesson-6-rule-of-thirds-part-1_21.html

https://www.jgrarchitect.com/2020/08/lesson-6-rule-of-thirds-part-2-serlio.html

 

https://www.jgrarchitect.com/2020/09/lesson-7-how-to-layout-frame-with-lines.html

 

A barn built in the 1830's

Green Mountain Timber Frames http://www.greenmountaintimberframes.com/ measured this barn before they dismantled it to use its frame anew.

Due to the wood used - poplar, beech, hemlock - the layout and the construction we think this barn was built by a farmer without an extensive background in framing. We think it dates to the  1830's.

 The floor was dirt, the head room under the hay loft not quite 6 ft.
What was it used for? Sheep perhaps? Sheds, windows and a silo were added over the last 180 years, making the original purpose hard to read.


I start with the farmer.
He had some wood of a certain size and length he could use for posts and beams for a barn. He knew how the barn would be used and where it would go.

Probably he had a carpenter square - they were readily available. But maybe not, as his dimensions don't quite fit. And he was much more comfortable with the old-fashioned geometry of the 'whirling square'.

He started with the width - 18 feet. He made a square: 18 ft wide, 18 ft. high - first diagram.\
Or so it seems. Today that height is 17'-10", 2". I think originally the width was also 17'-10". His inch seems to have been just a bit smaller than today's inch

He could have started with a string about 18 ft. long. He could have used a compass with a 27" radius, stepped it out twice for 4'-6", twice more for 9' and doubled that for 18'; or a pole 4'-6" long.

In his square he laid out his center lines and then the star that joins the points - the second and third diagrams. This is a medieval framing system which came to New England with the English colonists.

I have added circles to mark how the lines of the star cross at the locations of the girts, I've added a green dashed line to show how the height of the wall is 2/3 the height of the end wall. Almost. It's off by 2"

Using a carpenter square to layout a 3/4/5 triangle does not work as well. The wall height isn't high enough. The lower girt can be determined - see the green circles - but not the upper one.

While the frame appears governed by the traditional English framing geometry, the frame itself has dropped girts - a Dutch traditional way of framing. The girts are mortised into the posts below the upper beam. This combination of framing methods is sometimes referred to as 'American'.


The floor plan is simple: three 3/4/5 triangles. If the width is 18'  the length should be 40'-6" . It was 40'-2" measured on site. The men repairing the frame tell me it is 40'-1"; that the 2 interior bents are at 13'-4 1/2" from each end.

If one arm of the 3/4/5 triangle is 17'-10", the other is 13'-4 1/2".
3/4 of one side of a 17'-10" square = 13'-4 1/2". So either framing system fits the floor plan.

Dan McKeen, GM Timber Frames, also tell me 3 girts are beech, one poplar.The top plates are poplar and in good shape. The posts are sawn hemlock and hewn beech. The ties are sawn hemlock.

I looked at how did this farmer/framer laid out his girts in the side walls.
Here I tried - on the right in red - 3/4/5 triangles. The intersections - red circles - using a triangle that includes the rafter tails, are close, but convoluted. Not simple.
However, a square laid out inside the frame - on the left in green - neatly divides the space in thirds - green circles.

The star of the square used for the gable end laid out along the side also notes the placement of all the  girts.








The numbers on the early carpenter squares were engraved by hand. It is possible that this farmer/framer owned a square that was very slightly off. Or he used his own measure.

The man who built this frame comes alive as I study it; I've met him. Now I want to ask how he learned to frame - who taught him? what tools did he like? where did he start? were we right about his choice of materials?

The cobbler's house north of Boston, c. 1840. Part 4 of 4

Organizing my drawings and thoughts for  presentation has required me to revisit earlier ideas. So here is how I now think this simple house may have been designed.  




Here is a new post on this house, built north of Boston in the Merrimack watershed c. 1840. The previous posts include its history.





First: the geometry of the 3/4/5 triangle:

A triangle by definition has 3 legs. If the legs are in proportion to each other so that the shortest is 3 units, the middle one is 4 units and the longest is 5 units. the triangle will always be shaped like the diagram here - with the 2 shorter sides meeting at a 90*, the longer side always the hypotenuse.

A 90*, a right angle, is what carpenters and brick layers need - a   way to be sure they are erecting a stable shape that in construction will keep the loads directly over their bases and transfer those load to the ground.

The wide spread availability of carpenter squares after 1820 in the United States made the 3/4/5 triangle the easy and flexible choice for layout. 2 triangles with their hypotenuses side by side made a rectangle. The long and short sides of the triangle could be flipped.


Here in the floor plan of the  cobbler's house, The right side, the main house, is 2 rectangles made up of 3/4/5 triangles. (A) is the triangle laid out. (B) is 2 triangles turned - one starts from one corner, the other from the opposite. Where they cross is the center of the rectangle. The back wing is one 3/4/5 rectangle and a little more, The size of the 'little more' (C) is  determined by the 3/4/5 ratio.

The rectangles of the 3/4/5 triangle can be set side by side, as they are here, or they can be slide past each other as can be seen in the diagram.









Here is the side elevation of the cobbler's house - a 3/4/5 rectangle with 2 more for the roof - arranged much as was the floor plan of the house. (A) is the main box, (B) is the 2 triangles set back to back.

 Add the lines for the other triangles - the diagonals for the rectangle.Where the lines cross determines the placement of the 2nd floor beam and joist (dash and dotted lines above central 1st fl. window).
The intersections also mark the center of the house, thus where the 1st floor window goes, as well as the spacing between the 2nd floor windows.






The first drawing assumed the width of the house was the '4 units' of the triangle, that the height was '3 units'. If the height is seen as '4 units', the resulting triangles cross below the 1st fl. window, That space where they cross is the width of the window.

Here it is clear that the numerical length of the unit does not matter.
The relationship between the lengths, their ratio, is what determines the layout.





What happens on the front of the house?
The front elevation is made up of 2 squares. However, the layout of the door and windows does not come from those proportions.
It comes from the use of the 3/4/5 triangles - with the wall height '3 units'.  The 3/4/5 rectangles, begun on each end of the elevation cross in the middle. Their edges place the windows on either side of the door.

Lay out the rectangles (drawn here on the left side): the diagonals cross on the center of first windows.
Swing the triangle to its other leg (drawn here on the right side): the resulting rectangle's diagonals cross at the edge of the outer windows and mark where to place the window jambs of the inner windows.







January 2020: 5 years after I wrote this I think I should check the geometry once more. If the layout of the frame began at the sills  - as I have often found - the geometry is simpler.




Originally I thought that the geometry determined the design, I now think the geometry was used for the framing. The design came from the frame. These vernacular houses did not have an 'architect' but instead a master carpenter. While I am sure they thought about what a house looked like, they were builders, not draftsmen.
These houses resonate with us because we sense the geometry which relates all the pieces to the whole.

Eagle Square Manufacturing Co. - a history

 The University of Vermont holds in its collection the records for the Eagle Square Company of Shaftsbury, VT.  It is now digitized and available to researchers.

http://cdi.uvm.edu/findingaids/collection/eaglesquare.ead.xml

Its first paragraph clearly states that Silas Hawes was not the inventor of carpenter squares. Good. We need to lay that myth to rest.


When I became curious about what tools were used for the layout of buildings in the British Colonies and the early United States, information about them seemed non-existent.  It was hard even to find what library might be a place for me to go to research, if I had been free to go traveling for a week. I found a few cabinet maker's tool boxes: the Bennington Museum owns one. But it was not easy to find lists of  tools a master carpenter would have owned. It still isn't.

Information about the history of carpenter squares was part of what I wanted to know about. I still do. They were widely manufactured here, and  40 early squares are stored in the vaults of the Bennington Museum. But there is no repository of information about their use or popularity, just local stories like the one that says Hawes invented the squares.

So I am posting what resources I find here.

7/6/17: update. I have been reading Joseph Moxon's 1683 book which includes descriptions of tools and how to use them, carpenter squares among them. For more information read: http://www.jgrarchitect.com/2017/06/joseph-moxon-mechanick-exercises-london.html

Scribe Rule to Square Rule and Carpenter Squares

Here is a post on Will Traux's blog about the rapid transition from the use of Scribe to Square Rule in Timber Framing. (He explains what those different rules mean.) The comments are an important part of the discussion.

http://bridgewright.wordpress.com/2012/12/01/a-now-two-century-old-overnight-turn-on-a-paradigm/

I have also seen  the rapid change from one system to the other starting around 1820.

I thought this could partially explain why accurate carpenter squares became so popular after 1820, why so many mills sprung up along Paran Creek in North Bennington and Shaftsbury, Vermont, to manufacture the squares, why after being wiped out by the disastrous flood in 1852, the factories were quickly rebuilt.

The comments to the original post discuss the need for standardized dimensions when using the Square Rule system. I think they make a good argument that standard measurements were not essential.

So I haven't a answer, yet.

carpenter squares in 1503

Here are carpenter squares in use in the early 16th century in Germany.

The print comes from Robert Lawlor's Sacred Geometry, Philosophy and Practice, p. 7. The wood cut by Gregor Riesch is from his book Margarita Philosophical, published in Baden,1503. Lawlor's description reads:

"Geometry as a contemplative practice is personified by an elegant and refined woman, for geometry functions as an intuitive, synthesizing, creative yet exact activity of mind associated with the feminine principle. But when these geometric laws come to be applied in the technology of daily life they are represented by the rational masculine principle: contemplative geometry is transformed into practical geometry."

I add this to the posts on carpenter squares, geometry, and regulating lines to follow up on some ideas I'm thinking about.

a) Carpenter squares were in regular use centuries before Silas Hawes of Shaftsbury, Vermont, made his first steel square in 1815, beginning what became The Eagle Square Co. Local lore wants Hawes to be the inventor of the carpenter square, but it isn't so.

b)The practice of geometry has since its beginning been an intellectual, philosophical process - 'contemplative' in the words of Lawlor - as well as a practical skill called Practical Geometry.

c) The intertwining of design - contemplative and theoretical  - and practice in construction unraveled during the Industrial Revolution. As the master-carpenter and master-mason evolved into the separate professions of architect, engineer, and builder each lost parts of the knowledge and skills, as well as the understanding and appreciation of what the others was doing.

d) Notes added 1/7/2019:
The woodcut is a wonderful window into geometry, crafts, tools, even education in late Middle Ages.
The woman is using a compass,  a tool that historians often overlook.
The famous Geometers (Google it!) were men - some of our best thinkers: Plato, Archimedes, Pythagoras, etc., not 'contemplative' nor 'feminine' as Robert Lawlor writes. I would describe them excellent, theoretical, logical, careful thinkers who also understood how to use Geometry for Practical purposes..
The word 'feminine' has little relevance in a discussion about Theoretical and Practical Geometry.

Edward Shaw - uses the tools

This picture of a 1854 construction site had me hoping, even if it was idealized.

The architect - wearing the stove pipe hat - holds dividers as he measures something on the drawing for the observant and expectant carpenters. In the foreground on the grass is a carpenter square, a hammer, and a large compass.

Maybe Edward Shaw's pattern book, The Modern Architect, published in 1854, would mention geometry! Maybe I'd find mention of proportions in a paragraph about something else!

Well, he does say that a main floor window's height should not be more than double its width. Room length, breadth and height and height are mentioned in relationship to each other. But then he states that 10 ft is the desired height... There is great advice for the carpenter and homeowner about foundations, lath and plaster, and 'warming'. Fun, but not what I hoped for.

Shaw's life (1783-1859) spans the change from custom to repetitive parts in construction. The picture shows a building being balloon framed with 2x's , not posts and beams. The drawing in the illustration is being measured and scaled up by dividers, an ancient tool, not a modern architect's scale with regular increments. Almost anyone can draw circles with a compass. In the time Shaw practiced master carpenters and architects knew how to use compasses for design, layout and framing of rectangular buildings.

The book includes extensive explanation of how to lay out columns, scrolls for hand rails, and molding details that would require a hand held compass. The large compass shown would have been for stepping off foundations and wall locations based on the drawing made by the small compass. Or it is possible that the 'compass' is  perhaps a level, folded up.

The picture is the cover of the Dover Publications reprint of Shaw's book. Inside is a reprint of the etching in black and white. It is too dark to reproduce well. For a look at the original print try: http://www.historicnewengland.org/preservation/your-older-or-historic-home/articles/pdf149.pdf . It is part of a good article on a mid-19th century Maine builder in the SPNEA journal, 1967. SPNEA (Society for Preservation of New England Antiquities) is now Historic New England.

carpenter squares

In 1815, Silas Hawes in S. Shaftsbury, VT, joined 2 legs of steel together to make a stable, true 90* angle carpenter square. Hawes patented his idea in 1819 and began manufacturing. (Iron squares did exist before this. Illustrations of them can be found in the pyramids and in medieval English carvings. There was one recorded in Plymouth in the 1620's, and another in New Haven, CT, before 1700.)

I became curious about these steel squares when I realized that there were several factories producing steel squares on Paran Creek, which runs from Shaftsbury, through N. Bennington to the Walloomsac River. Lots of factories because of lots of demand - one factory, swept away in a flood in 1852, was immediately rebuilt.

At the same time Asher Benjamin is publishing his pattern books.
And post and beam framing systems are evolving from scribe rule to square rule. This is a change from each tendon fitting only one mortise, to the parts being interchangeable. For example, a brace could fit between the post and beam (sill and stud in the illustration) at the front of a barn or at the back.

Do these facts have anything in common?

A joiner needs to know the angle he uses will be the same each time, dependable, before he can make the same part to be used many places. He needs to own a carpenter square even if it is expensive, and it was - at least a week's pay.

Does the manufacturer of many, many carpenter squares in Vermont a play a role in the evolution away from design using 'regulating lines'?

The Eagle Square sign comes from The Shires of Bennington, published by the Bennington Museum in 1975. The illustrations were drawn by Edwin Tunis for his book, Colonial Craftsmen, the World Publishing Company, 1965.