Archive - Practical Geometry

Friday, February 20, 2026

The Practical Geometry of Boston's Old North Church

The Old North Church, in Boston's North End, was built in 1723.*

The church fathers responsible for its construction had visited England where they had probably seen the churches in London built after the Great Fire in 1669. Those churches were built using the new Classic architectural style which included half circle windows - as are in the church - instead of Gothic pointed ones.**

I visit the church when I teach/share/explore Practical Geometry with the North Bennet Street School Preservation Carpentry students. We, along with the instructors, see its construction first hand, from the crypt with its tombs to the timber roof frame in the attic.

This year I explored the church's geometry. The Historic Structures Report for the church, written c. 1950, includes floor plans which trace the evolution of the church interior. These are not HABS drawings, not as detailed and possibly not as accurate. This is, therefore, a preliminary exploration.

The exterior is brick. The floors and and roof are timber framed. One of the church fathers seems to have overseen the master mason responsible for the foundation and walls. A master timber framer was responsible for the roof. All of these men would have known Practical Geometry.***

Drawing #1 shows the church interior layout in 1730. The entrance and apse are extensions of the main space. The pulpit with its curvy stair is drawn to the left of the apse. In later years the pew locations, the stairs and some exterior doors change.

The geometry of the exterior walls is simple. It is the 3/4/5 rectangle. The 2 3/4/5 triangles and the rectangle are outlined in red.

In Drawing #2 the black diagonals of the Rule of Thirds cross the interior. Where they cross are the points**** which locate the red vertical lines which divide the space in thirds. These lines layout the inside width of the apse and the entry.

The apse plan is a half circle. The entry is square. Both are drawn in red. The church wall is a little thicker at the entry to support the steeple.

The distance of 2 of the columns from the back and/or the front walls is also set: see the dashed red line at the 2/3-1/3 division of the church interior.

Drawing #3: The square laid out in Drawing #2 by the Rule of Thirds can also be divided horizontally by the same Rule - see the black lines. The points allow 3 equal divisions across the church - the red lines, labeled A.

Drawing #4: By adding the Rule of Thirds' vertical lines - again the black lines - the points are found on the red lines (A). These points locate the columns on both sides of the body of the church: the vertical red line which crosses the A lines at B to mark the distance between the 3 columns on each side along the length of the church. The fourth column, located in the space at the bottom of the drawing, uses the same spacing as the other columns

I have drawn only the left side. The right side matches the left. Too many lines become visually confusion.

Drawing #5: The windows were centered between the columns.

In a time where candles and torches were the only artificial lighting available, placing the windows for maximum natural light was essential.

The arcs in black on the right side locate the center line (C) of the window.

The center of radii for the arcs is at the columns. IE: place your compass point on the edge of one column; swing an arc. Swing a similar arc from the opposite column. The line drawn by the 2 points where the arcs cross will locate the center of a window.

*The church is best known for its steeple where Paul Revere hug his lanterns on April 15, 1775, warning the citizens of Massachusetts Bay by which route the British soldiers would march to Lexington and Concord.

**This drawing is by Suzanne Carlson, from John Fitzhugh Millar's book, The Architects of the American Colonies, Barre Publishers, Barre, MA, 1968. Used with permission from John Fitzhugh Millar.

*** In 1723 someone who designed buildings was not called an architect. James Gibbs who designed and supervised the construction of St. Martin's in the Fields in London, was chosen by Commissioners to be "the Surveyor of that work" . See Gibbs' Book of Architecture, London, 1728, p. iv. There seems to be no use of that word in the church record.

**** Remember that in geometry, Practical or otherwise, a Line cannot be drawn without 2 points. To learn about the Rule of Thirds, see https://www.jgrarchitect.com/2020/08/lesson-6-rule-of-thirds-part-1_21.html

Tuesday, May 13, 2025

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