Teaching Practical Geometry

 Several educators, curious about Practical Geometry, have asked me how I would share this geometry in the classroom. This post is an introduction to how I would begin.

In September, 2023, I presented 3  workshops at IPTW, the International Preservation Trades Workshops.* The last day was open to the public. About 10 people, aged 10-70+, came to learn about Practical Geometry. Some had never held a compass.  

Here is what we did:

We drew circles with compasses. Then we divided the circumferences into 6 equal parts and connected the points to make rectangles and squares. We used no numbers. 

 

We  explored the  design and layout tools a carpenter would have had before the Industrial Revolution: the compass, a line and a scribe. We talked about how those tools were used and are still used. We compared cubits (the length from your elbow to your longest finger).  We set carpenter's dividers for a day's work by the radius or the diameter of a daisy wheel. One of the participants taught the others how to snap a chalk line.

I brought my daisy wheel with me. It was scribed into a 9 ft tall board which was once sheathing on Vermont barn, c.1780. The barn was deconstructed about 10 years ago. The deconstruction contractor gave me the board.

 

 

I showed them the floor plan of one of the early Virginia folk houses recorded by Henry Glassie,** which used the geometry we had drawn. 

 

 

 

I shared a few pictures including this house whose plan we had just laid out.  

 

 

That image introduced the class to the chimney wing. Its plan would have used the 3/4/5 rectangle to make sure the wing was parallel to the house so that all the roof framing could be cut the same length.  

 

 

 

I showed the group a Menagery, a retreat intended for an English gentleman's estate, designed by James Gibb's ***, c. 1720.  

 

 

 The wings are laid out in the same way, using the 3/4/5 rectangle. Here it is because the rough laid stone on the exterior would have made an accurate layout and construction difficult.

 

Then the class learned about the 'star', the Lines, in the center of the Menagery. Those are also the lines on our cellphones which help us edit images, known by artists as the Rule of Thirds.  

Here is the geometry: the diagonal of the square and the Lines from the ends of one side (the corners) to the middle of the opposite side. The  pattern is turned 4 times.

 

 

Where the lines cross are points. 2 points connected are a line. That line is always straight.

Here, the points divide the large square into 9 small squares - the diagram used on cellphones - or 3 equal rectangles.

There are also 4 squares within the large square. If their diagonals are drawn, the large square can be divided into 16 small squares or 4 equal rectangles.

 

The Lines on the elevation for this brick house tell the mason where the sides of the door and window openings are. On the plan the Lines show the fireplace edges and the placement of the interior walls. 

The drawing is Plate 56 in Owen Biddle's pattern book, The  Young Carpenter’s Assistant,  published in 1805, by Benjamin Johnson, Philadelphia.

 

 

 

 

 I ended with these Lines in Sebastiano Serlio's Book I, c. 1540. It explains where to place a door in a castle wall. He ends Book I: On Geometry, " However, honest reader, although the things resulting from the various intersections of lines is infinite, to avoid being long-winded I shall come to an end."

 

This was more than enough for one 75 minute session. 

Several shorter lessons would have been easier for everyone. There was very little time for questions, more examples, or in-depth understanding.  

 

For more information: In 2020,  I wrote 7 posts entitled 'Lessons' for students of all ages. https://www.jgrarchitect.com/2020/04/lessons.html .

*The 25th International Preservation Workshops were held this year in Frederick, MD, at the Hargett Farm which will become the Historic Preservation Trade Center for the National Park Service.          See the Preservation Trades Network website, ptn.org, for more information.

** Henry Glassie,. Folk Housing in Middle Virginia, U of Tennessee Press: Knoxville, 1979

*** James Gibbs,  Book on Architecture, London, 1728, Dover reprint

**** Sebastiano Serlio . On Architecture, Lyon, France 1530, translated in1611,  on-line and translated by Vaughan Hart and Peter Hicks, 1996, Yale University Press, New Haven

To read more about this diagram see https://www.jgrarchitect.com/2022/10/serlios-lines.html 

 

Stratford Hall, Part I: Paul Buchanan's Ideas

This post is an introduction to my understanding of the geometry of  Stratford Hall, the plantation and home of the Lees of Virginia for 4 generations  beginning in 1736.

Here is the house as it looked this spring when I was there for 3 days. 

In the gift shop I found the book  Paul Buchanan Stratford Hall.* Of course I bought it.

The book focuses on Buchanan's  research at Stratford Hall from 1984-1993, after it introduces us to his historic preservation work at Colonial Williamsburg as well as other Great Houses in Virginia, including Gunston Hall.

This image on page 14 is from one of Buchanan's favorite pattern books, The London Tradesman, by R. Campbell, published in London in 1747.

I spotted the compass leaning against the beam, and the man to the far left, probably the builder, with his rod. That led me to the book which is available online. 

It is a fascinating review of the trades in mid-18th c. London. Geometry is listed, among others, as a necessary skill. 

Chapter XXXI, Of Architect and those employ'd in that Branch, lists the skills men need to succeed in that profession.  Campbell writes that an architect's "head (must be) Mathematically and Geometrically turn'd."  and "Besides this Plan he generally forms a Model in Wood." pages155-6

For a stone mason, "Geometry is absolutely necessary". page158

For a carpenter: "He must understand as much Geometry as related to Measuration (the act of measuring) of Solids and Superficies (surface areas)". page 160

For a joiner: "His Business requires that he should be acquainted with Geometry and Measuration". page 161  

Of course I wanted to understand how Buchanan applied this, what he knew about the geometric skills listed by Campbell. I study how our ancestors used geometry as a practical tool for construction, as well as for design.

Using the HABS drawings for the house, done in 1969, Buchanan overlaid squares and a circle.

His squares 'work': they fit. However they do not tell us much about how to build the house. They do not provide clear and simple information - the layout of the plan, the size of the spaces and their relationship to each other - for the Master Builder who, at Stratford Hall, was William Walker.

 

I used the width of the wings as the dimension of a square to lay out the foundation plan using 5 squares. The middle square is centered between the 2 wings. The foundation would have been easy to layout with compasses, and a rod, both of which are in Campbell's illustration.

 

 

Twine, also essential, is not in the illustration. Maybe it was too skinny - just a line. (A deliberate pun: twine marks the 'Line' - as in 'chalk line'- which the builder needs in order to build.)

Here is the width of the east wing used as the radius for a circle, drawn in red. The 6 points of the circle locate the square, drawn here in black. 

I drew this for the readers of this post who might not know how a daisy wheel can be used. In the process I saw that the foundations for the fireplaces are also located, see the black dashed line.

For an introduction to the use of a daisy wheel see: https://www.jgrarchitect.com/2023/01/geometry-in-construction-practical.html

* Paul Buchanan Stratford Hall and Other Architectural Studies, copyright 1998,  Robert E. Lee Memorial Association, Inc. Stratford Hall Foundation, Stratford, VA.

 

 

Lesson 7, How to layout a frame with Lines

 Lesson 1 is here: https://www.jgrarchitect.com/2020/04/lessons.html

What's a Line?

A Line is a basic tool for layout and design.

Serlio, in 1520, used twine with a plumb bob as his Line. On  the frontispiece of his book, On Architecture, he drew his Line entangled in the metal scrolling. The end for tying is frayed.

 

 

 

 

Even through we now write 'line' without the capital L, we still use lines to keep our construction 'true' - straight, square, and plumb.

 

Patrick Kennedy, in Kentucky in 2020, used his line to keep his new stone wall straight. He tied his line to wood stakes. Rebar is also commonly used today.

photograph courtesy of P. Kennedy. 

Modern examples of using lines to keep construction square and plumb:
Checking a foundation's diagonals will ensure its rectangle is true. A square frame is easier to erect than one which is crooked.

A rectangle will have 4 square - 90* - corners if its diagonals  are equal. Here the far rectangle is not true. The diagonal is too long. It  should meet the corner at the upper spot.

Plumb lines ensure that walls are true or plumb: straight up and down, 

If an existing wall is not true the framer must know in order to compensate for the discrepancy. Here the plumb line with its  bob on the bottom shows that the foundation wall slopes.

How lines were used historically in construction can be seen in the frames of buildings built for utility. No one was showing off - just getting the job done - using what was ordinary.

Tuckahoe Plantation., Virginia

Cabins for enslaved people c. 1750.

 

 

This is minimal housing. We are lucky it survived.

 

 

 

 

 

Its geometry:

1)   A line - a length of twine - could have been used to layout the 2 square units.
Lesson 5 and Owen Biddle's Figure H show the steps. ,

 

The builder could have been laid out the foundation knowing only where to site the cabin and how big it would be. He didn't need drawings. He could set his foundation and lay the floor using only his Line and some pegs.

 

2 - The Lines measured the diagonals: if they matched the 2 squares were true. The diagonals also crossed at the center of each unit, locating the fireplaces, doors and windows.

 

The cabin floor could have became the framing floor. The Lines necessary for framing could be marked with a chalked line. 

Carpenters today build this way: they use the floor of whatever they are building as their framing floor. When the frame is ready it is lifted into place.

The square and its lines determine the wall height (red arrow). The 2nd floor is set at half the square. The roof begins as a square set above the wall. The roof pitch is the diagonals of the 2nd square. The chimney is 1-1/2 squares tall. 

 

 

 

 

 

No numbers were needed to lay out this cabin's frame, just practical geometry, some twine and something to mark the important points. The carpenters needed to understand practical geometry but not reading, writing or arithmetic.

 

 

 

 

The front elevation is an after thought. The plan and end elevation have already determined all locations.

 

 

Carpenters use center lines to locate windows and door. They then frame on either side of the opening. 

Masons, however, need to know where to end the row of bricks. The lines for brick or stone construction often end on the side of an opening.

 Owen Biddle's simple town house is a good example. It has 2 rooms up and down, plus a basement and entrance porch.

This layout could be drawn on the first floor just as the cabin could have been. A drawing might have been necessary for the front entrance with its fan light.  I have seen such drawings on wood which - when no longer needed - became roof sheathing.

Biddle's basic Lines locate the window location and height, the entrance size. 

His floor plan uses the Lines to locate walls and the fireplaces.  

The illustration was a guide for the 'carpenter assistant' so it is quite straightforward. It is the proportions that give the house presence.

Still, the house would be bare, spare as farmhouses are, without the details: the fanlight, the porch columns, the fretwork.

Other illustrations in his book use more complex geometries.

 

My introduction to the Tuckahoe Plantation cabin is here.\: https://www.jgrarchitect.com/2014/06/cabin-tuckahoe-plantation-goochland.html

When I teach Practical Geometry, we often lay out the Tuckahoe cabin with a line. The floor plan appears as we swing the arcs and mark the intersections.

Owen Biddle's town house is elegant, and very simple. A step by step analysis of the geometry is here: https://www.jgrarchitect.com/2015/11/owen-biddles-plan-and-elevation-for.html 

 

The previous posts in this series  Lessons 1-6  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

 

 

 

 

Lesson 6, Rule of Thirds, Part 2 of 2, Serlio

 

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 

 

 

 

 

 

 

 

 

 

 

 

 

Lesson 6: The Rule of Thirds, Part 1of 2

 

The Rule of Thirds is what artists call the grid that appears on your cell phone. It helps you compose and edit.

A variation of this is used in Practical Geometry. 

 

 

 

 

 

Sebastiano Serlio used this diagram in his book, On Architecture, published  in 1545.  He writes simple instructions for the reader; he says to construct the 'lines'. 

Note that the triangle (with its base at the bottom of the drawing) intersects the diagonals at the the upper corners of the door.  The width of the square is divided into thirds.  

Check how the division into thirds in the square above this drawing  lines up with those intersections.  Serlio is using a a variation of the Rule of Thirds.

 

 

Like Owen Biddle (see Lesson 5) Serlio sets out basic Geometry as used in construction in Book 1.

Then he explains how to solve problems.  He does not show how he knows where to draw the lines shown above. He assumes the reader knows. 

 Here are the instructions:

 

Draw a square;

Add the diagonals to your square. Where they cross in the center. You have point 1.  

 

 

 

 

 

 

Divide one side of your square in half. Now you have  points 1 and 2.
With 2 points you can draw a line.

 

 

 

 

 

      

Add diagonals in each new rectangle.            

 

 

 

 

 

 

 Add the diagonals from the square.                                            

If you were drawing this for a construction project on wood, on masonry, or on paper, you would not have separate squares.  All lines would be on your first square.  I have drawn each step without the extra lines for clarity.

Do you see that the center line does not pass through the intersection of the diagonals? If you were the builder you would know that your diagonals will match when the line in centered. In this diagram they don't. So you would move  your center line.

This is the diagram for Serlio's drawing for the door.

 

 For the Rule of Thirds (as we know it today) add the diagonals for the rectangles on both sides of the square.            

Note that you have intersections (4 points) not just where the lines  divide the square into smaller squares, but where the diagonals cross those lines.  2 points above the horizontal center line and 2 points below. Or: 2 on the right side of the vertical center line and 2 on the left.

I have deliberately not added black points where the lines cross. You who are reading this will see it more clearly if you find those points yourself. 

 

Connect those new points and extend the lines across the square. 
You have drawn the Rule of Thirds.

 

 

 

   

Similar diagonals could be drawn from the left to the right side and vice versa. 

 I drew all the diagonals on graph paper to make it easier to follow.  The next lines to add would be the diagonals of the small squares.
The line does not come back to its beginning until it has continued through the complete pattern. 

 

 

A post on Serlio. https://www.jgrarchitect.com/2017/04/serlio-writes-about-practical-geometry.html 

 

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 

 

Ruler & Compass, by Andrew Sutton

An excellent introduction to the "basic principles of geometric construction"!

A book I can easily recommend to a beginning geometer or an experienced one.


Ruler & Compass, Practical Geometric Constructions, Andrew Sutton, Bloomsbury USA, NY; U.S. edition, 2009.

It's part of The Wooden Books Series. The fly leaf says "An Introduction to Geometry without Measurements".
Andrew Sutton is a high school math teacher in the UK.

His illustration at the bottom of his dedication page:






This is a small book, 6" x7"  with 30 chapters, 58 pages. It includes sources, history, and many illustrations. It is dense, full of great details, but not intimidating.
He begins with an Introduction, Fundamentals, Perpendiculars, and Parallels.



These are his diagrams for his chapter (2 pages long)  "Squares & Rhombuses from lines and circles".

I had fun comparing Constructions 34, 35, and 36 to Asher Benjamin and Owen Biddle's instructions. Both 34 and 35 constructions seem easier and faster than theirs.




These constructions are also variations of ones  I've used.










At the end are an Appendix Polygon on Grid Construction and an appendix on Polygon Combinations.
This construction from page 56 is repeated and refined in 5 different ways.


The book refers to construction only as it is found in ancient Egypt and India. He does include diagrams by Serlio and Vignola, but seems to reference them through others, not from Serlio's and Vignola's own writings and diagrams.


I would like to hear his thoughts about Practical Geometry as it applies to construction.

Practical Geometry Lesson 5, Addendum

Why I left out diagram K from Owen Biddle's Plate 1 in his Young Carpenter's Assistant.

Lesson 5 was written for a student today who wants to draw rectangles using practical geometry.
Biddle was writing for the apprentices he worked with in 1805. They needed to know the practical application of geometry for the buildings they worked on - including the curved parts.

This addendum is like one of those long footnotes in an historic report -  a part of the story that's not quite germane to the subject, but ought to be included.


Biddle  identifies each diagram on Plate 1with a letter. There is no diagram for D. However, in his text, between C and E he discusses the mathematical instruments a carpenter should obtain. Perhaps this is D.  I quote him:

- scales of equal parts on the thin ivory or box rule
- a bow pen or compass
- a small piece of gum elastic for rubbing out black lead lines
- a stick of Indian ink
- 2 camel's hair pencils, one large, one small
- a black lead pencil



There is also no J. And there is no text in its place as exists for D. 




Here is K.  

Biddle writes: "Three points (not in a right line) or a small part of a circle being given to find a center which will describe a circle to pass through the points or complete the circle."

                                                     
                                                     Start with a curve a-b .
                                 The curve in Biddle's drawing above is a-b-c.                       










 The curve divided in half:  Swing 2 arcs that are the same length  above above and below the curve: a-c and b-d. Mark where they cross, at f above and below the curve,









Connect  f and f with a line - here dashed. Mark where the line crosses the arc a-b -  I've labeled it g.
This line divides the arc in half. 
If 2 lines were given - here: a-g and g-b , this step would not be necessary. Biddle's diagram  labels his lines a-b and b-c.

Now, the instructions become complex.
Draw it step at a time. And consider that this is only Plate 1 of Biddle's pattern book. He included 43 more Plates for the carpenter's assistant.  

Divide the lines a-g and g-b in half.
This is shown in Biddle's E  and F diagrams. Check Lesson 5. 

Extend the lines which divide  a-g and g-b in half so they intersect at k,
K is the center of the circle which passes the points or completes the circle.

Refer to Biddle's drawing K above for the complete solution, all neatly explained in only one diagram.  



Clearly Biddle thought this information  was essential knowledge for  every carpenter. His next Plates illustrate why. The construction his 'young carpenter's assistant' would be working on involved determining and laying out many curved lines for vaults, arches, windows, stairs and railings.

Plate 2 discusses ellipses: how to draw them using geometry or a trammel, how to find the center and axes of one already drawn.   















Plate 3 is concerned with octagons, arches, groins. the use of trammels, how to divide a line into parts.

I am quite fond of Figure 1, describing " an Octagon within a square." . Simple, quick, even obvious - if you know geometry.

I have seen  painstaking explanations of  how to lay out an octagonal using algebra: quite painful.

Plate 6 reviews raking cornices and "the sweep of a cornice which will bend around a circular wall and stand on a spring."

Plate 31 lays out "the section and elevation of a circular or geometrical stairs". Biddle includes in figure C  "the manner of drawing a bracket for the ends of the circular steps..." and the careful, detailed instructions.




Plates 32-35 - not included here - explain how to layout the newel, the falling moldings, the hand rail for such a stair.






Biddle's Young Carpenter's Assistant, Owen, Biddle, 1805, originally published by Benjamin Johnson, Philadelphia, and Roland and Loudon, New York. Reprint by Dover Publications, Inc. 2006. If you want this book, you can easily order it from them directly. It has an excellent 15 page introduction with bibliography by Bryan Clark Green.

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