Archive - 2026
Note: updated for clarity 9/26/2026
Serlio's cube: It's on the frontispiece of Sebastiano Serlio's On Architecture *, published c.1550, translated into English in 1611. The geometry shows the structure for adding a circle within a square - for example, a dome above a square space.
This lower section of the book's cover shows the tools a master builder would have used for design c. 1550. The cube is a layout tool.
Serlio's geometric diagram: a base of a Tuscan column. **
And his diagram for a Corinthian column. **
This is his drawing of a Roman temple; it is design and structure is based on the circle and its square.
He included the plan. For more details see my blog post on the temple ****
How can these layouts be drawn with a compass and a straight edge?
The first step is the geometry of the square. The diagram is in Audels' Carpenters and Builders Guide. published in 1928***.
Next: add the circles that are larger and/or smaller than the square. Below is a sequence.
- the square with its diagonals
- Half the length of the square's diagonal is the radius for a circle around the square.
- The mid-point of a side of the square gives the length of the radius for the circle within the square.
- The points where the inner circle crosses the square's diagonals determine a square within the first square.
Reader beware: This could go on ad-infinitum.
What if the beginning dimension is the diameter of a column? How can a square with the right proportions - the pedestal, for example - be drawn around that circle?
The diagram is from Edward Shaw's Civil Architecture* Plate 2. 'Problem X: drawing a tangent to a given circle'.
The radius of the circle (C-A) is half its diameter (D-E). By definition it makes a point at the circumference of the circle (A).
1) C is the center of the circle. C-A is its radius.
2) C'-A is also the length of the radius. A tangent to the circle, C'-A-C' is the length of the diameter. It can be 'trued' using step 3.
3) The diameter has a length, therefore a point at each end Both C's can be one end of a line which extends to the point C, (the center of the circle). If they are equal, the line C'-A-C' is true, sitting symmetrically on the circle's circumference.
4) Those lines, extended, create 2 more C'-A-C's. The length of those 2 lines is the diameter of the circle: 2 more points. The points can be joined to become the square around the circle.
5) C- C' becomes the radius for a circle around the square, around the circle.
Too many words! It's much easier just to draw it together.
I've written other blog posts about the use of this geometry in real construction.
Practical Geometry at MIT, April 9, 2018.
**** Serlio studies a Roman Temple, April 1, 2023
The Perfect Cube and its Sphere, July 16, 2024
James Gibbs' Steeples, February 25, 2022 This is squares, not circles.
* Audels Carpenters and Builders Guide, # 2, NYC, 1923, p.494
** Sebastiano Serlio, On Architecture, c. 1550; translated from Italian into English by Hart and HIcks,1996. illustrations on pages 125, 257, 341
*** Edward Shaw, Civil Architecture, 6th edition, published in 1852
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
