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 

  

Durer's alphabet via the book 'Good Eye'

A rift, not a book review

This is the cover for Good Eye, George R. Walker and Jim Tolpin's latest book about furniture design and proportion.*

The G comes from Albrecht Durer's alphabet in his  Instructions for Measuring with Compass and Ruler of his  Four Books on Measurement (Underweysung der Messung mit dem Zirckel und Richtschelettersyt, Book 3, published in Nuremberg, Germany, in 1525. 

Good Eye uses for Durer's letters for chapter headings. It discusses the design of the letters on pages 2-4. 

I last read Durer's book during Covid, 5 years ago. I wanted to understand Durer's knowledge and use of geometry.

Now I want to know more about his alphabet.

Durer used the numbers and letters of his time and place, the script of 1525 medieval Germany.The drawings I can find via the modern, online copies of his book are fuzzy.*  So his instructions are not clear to me.

 

I want to understand the basic geometric formation of the letters: Can I find out where did he begin? 

Here are Durer's letters A, B, C, D, F and Z.

 

 

 

I chose C and D.

I am laying out only the left hand C and D.

 

 

For both I began as Durer did, with a square and its center.  I have drawn these on graph paper to make the layouts easier to follow.

 

 

 

Using the arc of the square's side, here are the steps for dividing the square into 8 parts horizontally and vertically:

1) The square is divided into 4 smaller squares using a compass and straightedge.

2) the smaller squares is divided. 3) and again. 4) and again. That smallest width is 1/8 of the width of the square. The square can be divided into 8 equal sections both horizontally and vertically.

 

 

 

 

 

Here is the C. The circle within the square :

 

The second circle, its center moved one unit (1/8 of the width) to the right.

The circle cut at 1/8 of the width on the right side to create the letter C.

 

 

 

The letter D:        half the circle 

 

 

 

 

 

The second half circle with its center moved one unit to the left:

The leg of the D drawn one unit wide, set 2 units from the left side.

The serifs added: 1/4 of the circle whose radius is 1 unit wide. 

Look again at Durer's letter D to see the circles.

 

 

 

For more on Albecht Durer see: https://en.wikipedia.org/wiki/Albrecht_D%C3%BCrer  

*I have asked to borrow, via Inter Library Loan,  the 1977 translation into English of this book. It may be clearer. If so I will update this post after I've studied that.

    +                         +                          +                        +                      +                   +                        +

Good Eye is a good book for me, a geometer. It speaks from a different vantage point than mine. This is excellent. 

Here's why. The use of geometry in construction, including wood working, was passed down by master to apprentice, by hand.  I know of no teaching manuals for apprentices. From Vitruvius, (c.50 BCE,) to the 19th century pattern books, the writers assume the reader already knows how to use a compass, a scribe, and a line. We don't know what words they used to describe what they were teaching. We have almost lost the vocabulary as well as the skills.

Good Eye uses a different vocabulary to describe geometry. For example: the book uses focus for the center of a circle,  the point on which an arc pivots. I use Euclid's word: point. This is fine. I have watched George Walker explain geometry by sketching on a white board. I have seen Jim Tolpin's work and discussed it with him.  I know we are all exploring the use of geometry in design and construction. Good Eye helps me think more carefully. Thank you.

*Good Eye, George R. Walker and Jim Tolpin, Lost Art Press, Covington, Kentucky, 2024

William Pain's 'The Practical Builder' and Lavius Fillmore

 

 Here's the Old First Church, Bennington, Vermont.*

 

In 1803 the church elders invited the Master Builder, Lavius Fillmore, to build a new church in Bennington. He had already built 4 churches in Connecticut. The latest had been included in Asher Benjamin's first pattern book, The Country Builder's Assistant, published in 1797.**                               

Pattern books were architectural guide books for builders. Their images were studied and copied by gentleman scholars and master builders; their instructions studied and followed by apprentices, journey men, and carpenters.

William Pain, in London, had written many pattern books, 8 of which are known to have been available through book sellers and in private libraries in the States. While there is no written record of what pattern books Lavius Fillmore owned or might have seen, I think he must have studied Pain's The Practical Builder, printed in London in 1774.***

 

 

Here is the evidence:

 

 

This engraving, part of Plate XIV, The Practical Builder, explains the proper design for the 'Frontispiece of the Dorick Order'. Note the fanlight tracery.

 

 

 

 

 

 

 

 

 

 

 

 Compare Pain's tracery to that in the fanlight of the Old First Church. 
Fillmore has elaborated upon and refined Pain's design. ****

 

 The columns  however, do not match the illustration of a Doric frontispiece. They are topped by Ionic volutes. 

In this photograph they look like the ends of rolled up paper. Or maybe balls of white yarn?

 

 

This is Pain's 'Frontispiece of the Ionick  Order', part of Plate XVI. 

 

 

 

The volutes match those of the Old First Church.

 On the left side the 'entablature' (the section between the door frame and the roof) also matches that of the church.

 

 

 

Here is Pain's detail of the capital. The right side of the entablature matches the 2 sections, the 'architrave' and 'frieze' of the Old First Church door. 

The volutes on the columns in the church sanctuary also match those on the frontispiece. The columns also have the same architrave, frieze (the top part above the volutes) and the very top part with the dentils - the cornice -  as are shown in the drawing.

Notes:  

*For more about the Old First Church, see the church website: https://oldfirstchurchbenn.org/

** Asher Benjamin's first pattern book is available on line. The original can be read at the Historic Deerfield Library, Deerfield, Massachusetts. We know what books Benjamin studied; he copied their engravings and used them in his own books.

*** William Pain, The Practical Builder, or Workman's General Assistant, I Taylor, London, 1774, Dover Press reprint.

****I have drawn the practical geometry for the fanlight. See: 

https://www.jgrarchitect.com/2021/10/geometry-of-old-first-church-fanlight.html 

I also drew the geometry of the church, 10 years ago. It needs to be rewritten, made simpler and clearer.  

https://www.jgrarchitect.com/2014/11/old-first-church-and-daisy-wheel-part-3.html 

James Gibbs and the Rockingham Meeting House

This blog post assumes you, the reader, are familiar with James Gibbs' architecture. If you need an introduction or a review, check the end of this blog. You will see links to what I wrote about him and his work. See also Wikipedia.

 

Did anyone in the States study James Gibbs' books?

Yes. Gibbs' On Architecture*, published in 1723, was imported to the Colonies. We know the steeple designs were studied and copied**. 

His book, RULES for DRAWING the several PARTS of ARCHITECTURE*, was also in the Colonies. 

Both books were in bookstores and private libraries. 

 

Were the rules Gibbs drew standard knowledge? Or was he simply the first to write them down? 

Did builders follow his layout instructions?  

I don't know yet. I'm studying historic doors, leaving surrounds and architraves for later research.

 

HABS has measured drawings of the Rockingham Meetinghouse in Rockingham, Vermont. It was  built from 1787 to 1797.  The Master Builder was John Fuller. The Master Joiner - who would have built the doors - is not recorded. He could have been John Fuller.

I know the Meetinghouse well. I've studied it, given tours, taught and written about the geometry of its construction as well as how the door paneling fits by the Rule of Thirds.**

 

The main door

 

 

 

 

 

 

 

 

 The HABS drawing of this door

 

 

 

 

 

 

 

 

That drawing with the dimensions inked out in order to make James Gibbs' geometry easier to read.

2 squares.

The width divided into 6 parts, 3 noted. Then one part (1/6 the width of the door) determining the width of the surround.  

I have used the arcs and lines that Gibbs used for his door layouts. The radius of the arc is the width and height of the square. This is a builder's 'shorthand'.

This layout matches the door on the left in Gibbs' drawing shown above.

 

A line can be divided into 6 parts using the Rule of Thirds. See Part II of my post on James Gibbs and the Rockingham Meetinghouse. The link is at the bottom of this post.**

 

 The door for the right stair wing at the Rockingham Meetinghouse

 

 

 

 

 

 

 

 

 

 

The HABS drawing for the right stair wing door

 

 

 

 

 

 

 

 

 The geometry: 

2 squares and 1/6 added to the height ( the red rectangle at the top)

This geometry matches the layout of the middle door in Gibbs' drawing of 3 doors shown above.

 

 

 

 

Then, I tried using the 1/6 part of the door width  as a radius.
I placed 3 circles on the width, the red line across the middle of the door. The dimension of the circles is the radius x 2: simple geometry.

 Beginning at the bottom of the door I stepped off 8 semi-circles up  the right hand side. They are the same width as those across the width of the door. Those semi-circles lay out the height of the door surround, the beginning of the architrave and its height.

Finally, I saw that the width of the pilasters on each side of the door was the same width as the circles. See the circle on the left pilaster.

The HABS drawings are small. The dimensions were made to record the building, not to record the geometry. Either the recorder or I may have missed nuance. This year, when the Meeting House is accessible, I will measure the doors to see how close what I've drawn is to the actual doors.

 

*James Gibbs,  On Architecture, 1728, London, Dover Press reprint

                         Rules for Drawing the several Parts of Architecture, 1753 edition through the University of Notre Dame  https://www3.nd.edu › Gibbs-Park-folio-18

**   https://www.jgrarchitect.com/2021/12/james-gibbs-book-of-architecture.html

       http://www.jgrarchitect.com/2022/02/james-gibbs-steeples.html

       https://www.jgrarchitect.com/2014/04/rockingham-meetiinghouse-rockingham-vt.html 

      https://www.jgrarchitect.com/2024/05/how-to-layout-pediment-350-years-of.html

James Gibbs' Rules for Drawing the several Parts of Architecture

My previous post looked at Palladio Londinensis' instructions for the use of geometry to design of entrances.* I found that essential information necessary to the layouts was left out/ not understood/missing.  Given that background, reading and writing about this book by James Gibbs has been a pleasure.

 

RULES for DRAWING the several PARTS of ARCHITECTURE

IN A More exact and easy manner that has been heretofore practiced by which all FRACTIONS, in dividing the principal MEMBERS and their Parts, are avoided.

 
By JAMES GIBBS

The Third Edition, 

London      1753**

This is a small book, 28 pages of text, 64 engravings. Gibbs is simplifying the design of columns. He discusses the complexity of dividing a module (the diameter of given circle) into minutes and seconds; that it's difficult to "divide the small parts with a compasses" and may "occasion mistakes".

He starts, "Of Columns and their Measures". The heights of columns are listed:  "The Tuscan - 7 diameters. The Doric - 8 Diameters. The Ionic - 9 diameters. The Corinthian - 10 diameters. The Roman or Composite - 10 Diameters." Next he discusses Entablatures, then his 64 Plates.

PART I

I am curious about how did masons and carpenters working on ordinary vernacular buildings use Practical Geometry. Can Gibbs' engravings tell me about vernacular design c.1730-50? 

Here are Gibbs' notes on 6 doors.

 

Plate XXVII shows 3 door frames: Tuscan, Dorick, and Ionick.

Each door has a segmented line on the left side. The divisions start at the top of the base of the columns. The Tuscan and Dorick lines both have 5 sections, one of which is the entablature's height.

The Ionick door has 6 sections, one of which is the entablature.

 

Those sections are the modules for all the parts of the door. The module is a length, a diameter of a circle drawn by a compass. So how does builder choose how big to make it? Where does he begin? 

Gibbs writes, "First find the Diameter of the Column, give 6 Diameters from middle to middle of the Columns..." 

From that diameter comes the sizes: the spacing of the columns, the width and height of the door opening. The door frame is a 'semidiameter', half a diameter, a radius.

 

Gibb's drawings are spare, clean.  His explanation, The Ionick Door, Plate XXXVII, second paragraph, for "The Geometric Rule to find the height of the Pediment..."  is easy to follow. ***

 

Vernacular buildings in the Colonies had doors with similar entablatures. Do the entrances for the Rockingham, VT, Meetinghouse follow Gibbs' instructions?  I will check.

 

 

 

PART II 

Gibbs' Plate XLII,  'Three Doors with Archtraves'. 

Gibbs focuses on the architraves. I am looking at the doors. I want to know if our American builders use these rules to layout doors.****

The doors begin with a square whose length is the width of the door. The diameter of the square is divided into 6 parts. One part is the width of the frame, the Architrave, which today we would specify as the molding or trim. The middle door is taller: it adds one more part (1/6) to the width and height of its trim. The diagonal of both squares "gives the bigness of the pilaster upon which the Scroll is fixed."

The geometry for dividing a diameter - or any line - into 6 equal parts

Using your line as the length of the sides, draw a square. 1) Add the diagonals . 2) Add the center lines. 3) Add the 4 lines from corner to opposite center point. Note the points where the  lines intersect.  4) Connect those points with lines.

You have divided the square into 3 long rectangles, and your line into 3 equal parts. See '1/3,1/3/1/3' above the square.

The distance between the center line of the square and the closest vertical line is 1/6 of your line . See "1/6" below the square, lower right. 

                                                                                                                  

 

*  https://www.jgrarchitect.com/2024/12/palladio-londiensis-frontispieces-c1755.html

**I am reading this through the University of Notre Dame  https://www3.nd.edu › Gibbs-Park-folio-18 

The first edition was published in 1732. It was available for purchase in the Colonies. I am always interested to see what words and phrases are capitalize in books printed in this era. 

***For  more information about pediments see my posts about Vignola's Rule for Pediments 

**** Today, a builder has a catalogue of doors to choose from. The doors may look different, but their widths and height are  similar: exterior doors are 3' x 6'8", 3'x7'. Other sizes must be special ordered or custom-made.  Before the Industrial Revolution there was no such uniformity.

 

Palladio Londinensis' Frontispieces, c.1755

:

Reader beware: This post is a work in progress.  I thought this was a simple post: I wanted to share the 'Diameters', because visually, with no words, they show how Practical Geometry was used.   

However, I am also reading James Gibbs' Rules for Drawing the Several Parts of Architecture..." * I found I was comparing of English pattern books c.1755. It was too much for one post. I edited. 

This post is about Salmon's illustrations. I've written a companion piece using James Gibbs' engravings.

These engravings are from Palladio Londinensis,The London Art of Building,  a builder's manual produced and published by William Salmon from 1734 to 1755.  Salmon was a carpenter and builder northeast of London. His book was readily available in bookshops and libraries in Britain and  the Colonies from the 1750's into the early 1800's.**

 

 

William Salmon's  Composite Order, Plate XXVI 

He wrote "..the Height of the Door is 3 Diameters, and hath a manner of pannelling different from all the foregoing; also the Entablature is 1/5 the length of the Pilaster, as may be seen from the Circles." 

The diameters Salmon mentions have no numerical value, nor do the Circles. They are the proportions for the door: 1wide/3 tall, and then for the pilasters and the entablature (the part of the door frame above the door itself and below the pediment): 5/1.  

A builder would have known generally how much space - width and height - he had to work with.  The circles and the diameters (semi-circles) were units of measure, the 'module' for that door's layout and design. Stepping off the module and adjusting its length, ie: its diameter, to fit the space he had, the builder could find the actual lengths of the door, its surround, and the entablature.** The layout of the frontispiece, the piece at the top, could come later. 

 

Before standardize dimensions, lengths were 'stepped off' using a compass. The diameter is the visual symbol of the compass' span and the act of swinging it. Salmon's book included drawings of diameters in rows - 'stepped off'. 

This illustration, from a 1950's text book for technical drawing, shows the compass stepping off 3 times.  

 

 

 The 'Doric Order Frontispiece and Door, Plate XXI'.

The height of the entablature is set by the length of the pilasters. The diameters on the right side, the modules, divide the length of the pilasters into 4 parts. 1 more part is added for the entablature.

The pediment is laid out by Vignola's Rule**  

 Salmon gives no geometric relationship between the door's proportions and the pilasters.

Is the circle drawn on the door its module? 
The door's length is 2 large circles plus a small one. The upper large circle encloses 4 small circles, so the whole door is 9 small circles tall. 

How would a carpenter find the diameter for the smaller circle if he began with the large one? It can be done, but not easily or quickly. Using the small circle as the module would be easier.

 The geometry:  Lay out 4 circles on a line. The length of the line from the outside edge of the first to the outside edge of the last is the large circle's diameter.

The point where the 2nd and 3rd circles cross the line is the center of the large circle. The circle's radius is the length of the line from the center to the outside edge of the first small circle.

Using his compass the builder could step off 4 small circles, or one big one for the width of the Dorick door, and 9 small circles for its height.

 

This 'Corinthian Order, Plate XV' door is 2-1/2 circles. The surround is 5 circles, the entablature is 1. 

The geometry of the door,  2-1/2 circles does not determine the size of the lower panel, the upper opening, or the size of the panes of glass. Salmon doesn't seem to understand that these sizes could have been derived from the proportions of the door itself. 

There is no indication that the pilasters' width or height might have been chosen to be in proportion to the door, or vice versa.

 

 

 

'A Dorick Entrance, Plate XXII' is an arched entry without a door, 5 modules long and, as the circle tells us, 3 modules wide.

The capitals of the pilasters around the arch are located at the center of the big circle. But the columns on each side of the opening and their pedestals do not use the proportions of the entry's 5 small circles.   

 

 

Here is the very simple geometry: the 3 circles.

 

 

Since each was drawn with a compass, each has a center. Therefore the big circle which encloses them is easy to draw; it has a radius  of 1-1/2 little circles.

 

London and the Colonies in the 18th C. needed builders. Some were well trained; others not quite.  Along with instructions for laying out Entrances, William Salmon's book included chapters on 'Geometric Problems', 'Prices of the Labor and Materials' for the trades as well as 'all sorts of Iron Work', information about staircases, lumber, roof framing, 'Chimnies', and the 5 Orders of Columns. 

The polite conclusion is  that Palladio Londinensis helped builders educate themselves, that in spite of its shortcomings it was a useful reference. Even so, I find his explanations inadequate and sloppy.

 

* James Gibbs, Rules for Drawing the Several Parts of Architecture..., London, 1753. The print and drawings are clearer and much easier to read online. 

**My copy:  Salmon, William, Palladio Londinensis,  London, 1755, Gale Ecco reprint. The original is often found in historic libraries. One is in the library of Gunston Hall, in Virginia, .

***Today doors and their surrounds come in standard sizes. Before the Industrial Revolution, a door and its frontispiece might be match another next door, or not. 

**** See my blog posts about Andeas Palladio's 'module',  https://www.jgrarchitect.com/2024/05/a-daisy-wheel-is-module.html, and https://www.jgrarchitect.com/2020/04/practical-geometry-lesson-3.html

***** Yes, I wrote a post about that too: https://www.jgrarchitect.com/2024/05/how-to-layout-pediment-350-years-of.html