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 

 

Practical Geometry, Drawing the Diagrams #2, the 3/4/5 Triangle

Here's the second diagram I taught at the 2016 PTN Workshops.

I did not lay it out as I have done here. Today I think this diagram would have been a good handout.I could have drawn it; the participants could have followed along and had a cheat sheet to take home.

4/18/2017: this diagram is awkward. I will redraw and simplify.

Using the  3/4/5 triangle for construction

 3/4 5 triangles always have a 90* angle where the side with 3 units meets the side with 4 units.

Draw a line and mark off your unit.



Lay out lines of 3 units, 4 units and 5 units.
On my diagram:  A-B = 3 units
                            A-C = 4 units
                            B-D = 5 units

Swing an arc from either end of A-B; one arc with a 4 unit radius, one arc with a 5 unit radius,
Where the arcs cross is E.





Draw lines from A to B  to E to A.
This is a 3/4/5 triangle. The corner at A is 90*

For fun I have laid out another triangle beginning with 5 units, use 3 and 4 units for the radii of the arcs. It is another 3/4/5 triangle with a 90* corner.

We used Gunston Hall, built of brick by George Mason from 1755 to 1759, as an example. Mason  was a real mason; he gave George Washington advice about mortar recipes. He would have used the 3/4/5 triangle when he built walls or, as a Master Mason, instructed others. The triangle was/is one way to keep brick square and true.
It would have been ordinary for him to use 3/4/5 geometry to design his house.

The base of the brick work at Gunston Hall is 4 units. The height of the brick work of the end wall at Gunston Hall measures 3 units. The diagonal is 5 units.

The floor plan is also laid put using the 3/4/5 triangle. See my post for more information and drawings:  http://www.jgrarchitect.com/2014/05/gunston-hall-ason-neck-virginia.html

I asked the participants at the PTN session to divide the width of the Gunston Hall side elevation into 4 equal parts. I wanted them to draw the geometry for themselves, to see it come to life.

Again a handout with step by step instructions would have been helpful.   

Not everyone knew how to divide a line into parts; but those who did showed those who didn't. It was a excellent group.

One of the first figures in the pattern books on Practical Geometry is the division of a line by a perpendicular. Here is  Figure 5, Plate II, of  Asher Benjamin's The American Builder's Companion, first edition published 1806.











 Asher Benjamin's Figure 3, Plate II,  shows a
simply drawn 3/4/5 triangle expressed with units 6/8/10
with short arc lines at c, the top, to show the use of a compass to make a circle with the radius determined.

His description assumes a familiarity with the language of geometry and compasses.

"To make a perpendicular with a 10 foot rod. Let b a be 6 feet; take eight feet in your compasses; from b make the arch c, with the distance ten feet from a; make the intersection at c, and draw the perpendicular, c b. "

Practical Geometry - drawing the diagrams, #1

The participants at the hands-on sessions I taught on Practical Geometry  at the 2016 PTN Workshops, asked me to post the diagrams for the basic geometries they worked with.
Here is the first.

How to divide a square into thirds:

We used graph paper for the first geometry so everyone could see the lines develope into a pattern. Everyone could count the squares to be sure they were following directions.

1.  Draw a square 12 units wide and 12 units long. label the corners A, B, C, D.

Add the diagonals - the lines from one corner cross the center to the far corner. A to C; B to D.
The lines will cross in the center of the square. Count the units to prove this to yourself.
Label the center of the square E.

Divide the square in half vertically, F to H - follow the line the graph paper.
Divide the square in half horizontally, G to I - follow the line on the graph paper.

This is the basic pattern. The square can now be divided into 3, 4, or 5 (or more)  equal rectangles as needed.


2. To divide the square into thirds:
Add a line from each corner to the middle of the opposite side. A to G and B to I.
These lines cross the original pattern.

K and L, if extended parallel to A-B, would define a rectangle that is 1/3 of the whole square.




A rule in geometry is that there must be 2 points to establish a line.
 Below is a diagram of  how the diagonals from the corner of a square to the middle of the opposite side give 2 points for the lines which divide a square into three rectangles of equal size.



This division of the square into thirds is often found in pre-Industrial Revolution design.
I do not think framers drew out the whole diagram on a sheathing board or a framing floor. Rather because the diagram was common knowledge they just drew the parts they needed.


An example:

At the workshop I taught the application of this geometric pattern using the plans and elevations from a cabin at Tuckahoe  -   http://www.jgrarchitect.com/2014/06/cabin-tuckahoe-plantation-goochland.html.

 

 

The end wall of the cabin is  2/3 of a square. The roof  begins on the 2/3 line. Its pitch follows the diagonals of the upper square. the windows, doors and fireplace are centered on the square. That's all.

I then showed the group how Owen Biddle used the same geometry to tell a mason where windows and doors were to be placed.
The  elevation and floor plan are both composed of 2 squares. On both the window placement is one side of the center line. The  diagonals from corner to center call out the window width ( on the elevation and the interior partition on the floor plan.
In the floor plan I have used a dashed line to note the lines dictating the window width.  
http://www.jgrarchitect.com/2015/11/owen-biddles-plan-and-elevation-for.html
 

 

 

I did not include diagrams showing how master joiners laid out squares bounded by circles, bounded by squares to set out the dimensions and relationships  between parts of doors and architraves for Georgian meeting houses.
Shown here is how the surround of the main door for Rockingham Meeting House, Rockingham, VT, may have been laid out.

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




The carpenters and masons called these geometries 'lines' They would have have been explained verbally as a master taught an apprentice.  Sebastiano Serlio and James Gibbs both refer to 'lines' - see my post  http://www.jgrarchitect.com/2017/04/serlio-writes-about-practical-geometry.html

Practical Geometry - as described by those who used it, Part 2

The last post  discussed how Asher Benjamin and Owen Biddle presented Practical Geometry in their pattern books in 1805 and 1806.
This post focuses on Minard Lefever, and finally Peter Nicholson, who inspired them all.

Minard Lefever ( 1798-1854) wrote 5 pattern books between 1829 and 1856.
The Modern Builder's Guide was published in September 1833, in New York.
In his Preface Lefever says "...it will be proper to specify the authors whom I have either consulted or made extractions from,..."
One of these was Peter Nicholson.  Because Lefever copies Nicholson's drawings  directly I will post only the latter's introductory geometry.

Lefever writes 35 pages of  descriptions for 21 plates on "Geometry Adapted to Practical Carpentry".
Here are Plate 8  and Plate 20.


















Minard Lefever, The Modern Builder's Guide, NY, 1833, reprint by Dover Publications, NY, 1969.






Peter Nicholson (1765-1844) practiced architecture, mathematics, and engineering in Scotland.  He taught and wrote 27 books.  The Carpenter's New Guide was first  published in 1792 in Great Britain. His books were regularly reprinted in the States.

The book reproduced here was printed in Philadelphia in 1830, his 10th Edition with, he writes,"6 new Plates".  The book is 121 pages long not including the Index.
27 of those pages are of - as his title page says - Practical Geometry for Carpentry and Joinery, "the whole founded on the geometric principals; the theory and practice well explained and fully exemplified" on 10 copper-plates.

In the Preface he says, "...it is Geometry which lays down all the first principals of building, measures lines, angles, and solids, and gives rules for describing the various kinds of figures used in buildings; therefore, as a necessary introduction to the art treated of, I have first laid down, and explained in the terms of workmen, such problems of Geometry as are absolutely prerequisite to the well understanding and putting into practice the necessary lines for Carpentry."

His introductory geometry plates match those of Asher Benjamin, Owen Biddle and Minard Lefever, all of whom acknowledge him in their prefaces.

Nicholson's Plate 10 is Lefever's Plate 8.

I will bring this book to the 2016 IPTN Workshops in September. It is fragile.

If you would like to read the titles of Peter Nicholson's books, they are listed at the end of his Wikipedia biography.

Other architectural historians must have looked at the first pages of these books. Everyone cannot have just turned to the illustrations of mantles and window casings, building plans and elevations and ignored the plates on geometry. Why hasn't someone else wondered out loud why so many pages on geometry were included in a book about construction?

Someone must have considered that if Nicholson's The Carpenter's New Guide went through 10 editions and was published in the States as well as Great Britain - as well as being directly copied - that carpenters were reading it, using it, that his information was useful, that maybe we should understand what he wrote.

The builders who came before us used geometry to design and build. The knowledge was taught to the next generation hands-on. Books were not needed.
Boys were 'apprenticed', learned their craft and became 'journeymen', traveling to sites to earn and learn. Eventually these men became full carpenters, 'masters', and were admitted to a guild. The guild system was not always possible in the States. Men quit their apprenticeships. moved west or into cities. The skills and knowledge that masters were expected to impart had to be taught in other ways. Asher Benjamin and others set up a school in Boston. The pattern book was another solution - a way for 'young carpenters'  (to quote Owen Biddle) to teach themselves the necessary construction skills, beginning with geometry.

Part 1 can be read here: http://www.jgrarchitect.com/2016/08/practical-geometry-as-described-by.html

Practical Geometry - as described by those who used it, Part 1

Herein:
Asher Benjamin, Owen Biddle, Peter Nicholson, and Minard Lefever

What they wrote about Practical Geometry in their pattern books: Asher Benjamin in 1806, Owen Biddle in 1805, Peter Nicholson beginning in 1792, Minard Lefever in 1833.

I want their words to be easily available to anyone who is curious - someone who comes upon this blog or someone who comes to the 2016 IPTN workshops in September.

Remember that the pictures can be expanded - click on them.



Asher Benjamin's,The American Builder's Companion, was first published in 1806, updated and edited through 6 editions to 1827.


His title included the various chapters he has included. The first is  "Practical Geometry".

In his preface he says, " I have first laid down and explained such problems of Geometry, as are absolutely necessary to the well understanding of the subject."


His first 18 of 114 pages are about using geometry to design and build.



I have copied here his Plate I and its accompanying notes.

Asher Benjamin, The American Builder's Companion, Boston, MA, 6th ( 1827) edition, Dover Publications Reprint, 1969. Benjamin wrote at least 6 pattern books beginning in 1797, all popular.


Owen Biddle's book , Biddle's Young Carpenter's Assistant, 1805, was half the size of Benjamin's, easy to tuck into a tool chest. His first 9 pages of 112 are devoted to Geometry.
First comes how to construct a drafting board and attach paper to it, followed by how to make a T square and what kind of instruments to use. Then he says, " I shall now proceed to explain some of the most useful geometrical problems, which every Carpenter ought to be acquainted with". p.4

Owen Biddle, Biddle's Young Carpenter's Assistant, Philadelphia and New York, 1805, Dover Publications reprint, 2006. This is his only book  A respected master carpenter in Philadelphia, he died in 1806.

to be continued....

2016 Preservation Trades Network Workshops, September 9-11, Clermont Farm, Berryville, Virginia

The annual gathering will be at Clermont  Farm now owned by the Commonwealth of Virginia.

Here is the link to the farm: http://www.clermontfarm.org/ Their facebook page has good pictures.
The National Barn Alliance will be there too.  On Friday there will be a barn tour - http://ptn.org/iptw-2016/barn-tour

There will be blacksmiths, lime mortar makers, timber framers, window repair people, masonry specialists, painters, roofers...


 Last year at Shelburne Farm I watched dimensional lumber come out of a log with bark, all by hand. I saw a Georgian cabinet built, and windows become like new.  The pictures are from that gathering.













I will be there to teach 2 sessions on
                Practical Geometry
 which, to quote Owen Biddle in 1805 "every Carpenter ought to be acquainted with".

Or more formally: "Geometry is the foundation on which practical Carpentry is based." Minard Lefever, 1833,

The sessions will be hands-on.
I will have compasses, pencils, erasers and straight edges. And drawings.
I will be helping whoever shows up see the geometry which governed framing and design for churches, mansions, houses, barns. As we uncover the geometry participants will see how design and structure come from the compass.

We will decipher brick houses in Virginia, wood frame churches in New England, houses built from 1680 to 1840. For people who want to see how much they already know I will have the plates from the first pages of the pattern books which present  "such problems in Geometry, as are absolutely necessary to the well understanding of the subject." (Asher Benjamin, 1827) Will they master the problems with a compass and a straight edge?

The pattern books of Asher Benjamin, Owen Biddle, Peter Nicholson, Minard Lefever,  will be available along with posters and handouts on Robert Adam and William Buckland.
And paper for experimenting

I demonstrate twice. There will be  plenty of opportuity for me to watch and learn from the other presenters, to explore the farm and its buildings, and talk with people. I know I will have a great time.

You can come too.

Learning from a Workshop

The IPN Workshops at the Shelburne Farms Coach Barn were superb.
The barn is magnificent. To be able to be in and around it for 4 days was true luxury.

Above left is the main entrance from the court yard. Right is the dormer for the hay loft door above the stable. Below is half of one barn door showing its hinges and brick work.



The food was plentiful and excellent - local and fresh.
The company and the workshops couldn't be beat.
Of course I plan to go to the 20th annual IPT Workshop to be held in Virginia, autumn of 2016 . www.ptn.org


My presentation was almost derailed by the cheap school compasses I brought. The compasses did not hold their angle, so the diagrams we drew weren't true. I had not anticipated that the participants might not know how to draft: they needed basic instructions and better tools.

Luckily people bore with me and I presented twice. Many people talked with me about geometry between sessions.




Here is what worked best:

The daisy wheel: As people found the rectangle created by 4 points  they easily understood the geometry of the Old First Church in Bennington.








Making a square:  beginning with a line and a circle.

The hardest part for people to figure out was how to draw the arcs for the vertical line. I felt very successful when I heard one person explaining how to do it to another.










The 1830's farmer's cottage pleased everyone. They could see how to use what they had drawn.


A few people were able to rotate the square 45* to complete the diagram as shown

I brought the pictures from my post on  Asher Benjamin, Owen Biddle and Peter Nicholson. http://www.jgrarchitect.com/2015/07/geometry-of-cobb-hepburn-house-part-2.html

 We were to draw squares based on their diagrams  - as shown here:
This is where, especially, the compasses were not up to the task. The squares were not true;they were cock-eyed. I loaned my good compass out - so much easier to draw with good tools!
So, I explained and demonstrated. People practiced.
The pattern books' first pages of geometry turned out to be an adventure.


I thought to show 4 different ways to grow a layout from one dimension.
Instead I used the different buildings and diagrams as illustrations as people asked questions,
Good, thoughtful questions.

What a good time we had!