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 sheathing board of a c.1780 barn in Danby, VT, which 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 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.

 

 

 

 

 

These Lines were used by Sebastiano Serlio**** in 1540, when he drew this illustration. It explains where to place a door in a castle wall.

 

 

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

 

 

 

 

 

  

 

Stratford Hall, Part IV: Placing the Windows

Updated 8/26/2023

In the previous post* I wrote about all the construction details of Stratford Hall which William Walker, the Master Builder, had to work out before the masons could begin laying the walls.

They are all visible in this photograph: the brick pattern (Flemish Bond) and its variations, the brick headers over the windows in the ground floor and the main floor, the corner brick pattern and that around the windows, the cap at the shoulder, where the walls become thinner by one brick.

The Hall needed big windows on the main (family) floor above, smaller ones on the ground (service) floor below. As William Walker placed and sized the windows he would have considered all these constraints.

I wondered:

Was he drawing on paper, a board, a plaster wall, a framing floor? 

Drawings about framing can still be seen on cathedral walls and floors. So perhaps Walker did his layouts on the floor of the Hall.

He could have used the ground floor as a framing floor as soon as the foundation was set. The floor itself, his sketches and calculations, would become covered with pavers when the Hall was finished and ready for use.

It would also have been easier to lay out his ideas, check his dimensions for both floors from the inside, rather than working on scaffolding outside or in a shed nearby.

Walker used 5 equally sized squares to lay out Stratford Hall's floor plan. Perhaps he used a similar simplicity for the elevations. I drew squares.  

The red boxes show the interior of the wing. My tentative pencil marks are barely visible.

 

Simple squares (using the room height as the length) lay out the window locations for the Hall. Here is the main floor wall with 2 squares. They mark the edges of the center window for the main floor and the ground floor below.

 

 

They also locate the center of the window. The window frame is 2 squares tall - as noted by the diagonal line.

The size of the main floor windows and the width of the ground floor windows is set.

Squares of the same proportions, moved to the sides of the window, locate the outer edges of the windows on the left and right. Note: Square A-A and Square B-B.  

Those windows will be the same size as the center windows. There is plenty of room for the flat arches above the windows; the edging brick patterns are not crowded.

 

The only unknown is the vertical height of the ground floor windows and how far above the floor they will sit. Space must be given for the rowlock arches over the windows.

The Lines which located and sized the main floor windows extend to the ground floor. I've labeled them A-B, B-A, A-B to match what I drew earlier. 

Using the window width as a radius, and the floor of the lower level as the base, (the Line below 3) the center of the circle can be found, the circle and its daisy wheel drawn. It marks the brick arch over the window. It also intersects the Lines of the window widths  locating the height of the ground floor windows. 

The windows are 3/4/5 rectangles.

Walker trained as a joiner in Scotland at the time when James Gibbs, also a Scotsman, was there, and when Gibb's book, 'On Architecture', was published. Gibbs' book included plans for 2 'menageries'.** One was laid out with similar crossed squares, the other used the inside for its geometry as the exterior wall - shown here - was irregular, as is the exterior surface at Stratford Hall. Gibbs also used the 3/4/5 rectangle for layouts.

Copies of James Gibbs' book came to the Colonies. It is possible both William Walker and the Lee family had read the book in Virginia.

* Previous posts:

https://www.jgrarchitect.com/2023/06/stratford-hall-and-paul-buchanan.html

https://www.jgrarchitect.com/2023/07/stratford-hall-part-ii-geometry-of.html

https://www.jgrarchitect.com/2023/08/stratford-hall-part-iii-look-at.html

** My posts on Gibbs' geometry:

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

https://www.jgrarchitect.com/2022/01/james-gibbs-of-architecture-draughts.html

*  See previous posts:

https://www.jgrarchitect.com/2023/06/stratford-hall-and-paul-buchanan.html

https://www.jgrarchitect.com/2023/07/stratford-hall-part-ii-geometry-of.html 

**Personal note: I have been asked at workshops how I use practical geometry. Here is an example of how I would approach a design today. The geometry would tell me what size the windows would be and their spacing.

Stratford Hall - Part III: Designing with Brick

 

I had just published 2 posts on the practical geometry used to frame/construct Stratford Hall.*

The thinking, drawing, and writing about the overall structure of the Hall was done. 

I began exploring the drawings. 

The window layout was rhythmic, straight forward: all the windows on each floor were the same size. The space between them as well as the spaces between them and the corners were equal.  

 

As an architect I have used this pattern on new wings for old houses. I would size the windows to compliment the existing; then divide the remaining space evenly. Or perhaps I would give a little more width to the outside walls, depending on room layout requirements.**

Here, I thought, is a similar layout. 

However,  I was wrong.

As I thought about the proportions I was laying out a wood frame:  locating posts, beams, center lines of windows, heights indicated for headers, a window schedule. I was building the structure. The character of the exterior and interior would come later with the addition of sheathing, siding, and plaster finished with trim and moldings. 

The interior and the exterior of a brick frame are the structure, they are not added later. Stratford Hall was built of brick. Its character comes from the manipulation of the brick.

So, from the beginning the masons who worked for William Walker, the Master Builder, needed to know:  

 

Where to lay Flemish Bond using brick of just one color? Where use the whiter bricks as headers?

Where to leave window openings? How wide, how tall? 

 

 

 

What brick pattern to use at the window openings? How many bricks? At the corners of the wings too?

Where to use a flat arch as a header? Where a rowlock arch? 

Where to place the brick cap and make the brick wall thinner?

That question was easy to answer: where the wall diminishes just above the joist pockets which hold the main floor joists.

 

 

In this image you can see how much information William Walker had to give the masons as they began. 

How many rows at grade before the  
the corner brick pattern begins, as well as the use of lighter headers?  

What is the height of the ground floor windows, and their width?  

The brick details at the sides windows and at the corner which needed careful calculations are visible here.             

The black line at the corner is an electrical cord.

 

When I knew the issues I could ask the question:  

How did William Walker, Master Builder, lay out the elevations?

 

I will propose some answers in the next post.

*  See previous posts:

https://www.jgrarchitect.com/2023/06/stratford-hall-and-paul-buchanan.html

https://www.jgrarchitect.com/2023/07/stratford-hall-part-ii-geometry-of.html 

** I knew about the use of proportion and rhythm in architecture and applied it for many years before I 'discovered' Practical Geometry. I studied the illustration in pattern books without understanding the numbers engraved along the sides of those drawings nor pausing to read the texts. I did not 'discover' the practical use of geometry: I finally read the text in the books and studied the scales.

 

 

 

 

The Geometry of a Corn Crib

Our corn crib:  built c. 1810, photographed c. 2000. 

Well used, in need of attention, like every building on the farm.

 

 

In time we replaced the roof. Then we removed rotten siding.

 

 

 

  

 

We were surprised!The crib's post and beam frame was scribed! It has neat, precise marriage marks which, in this part of Vermont, are found on frames which were built 1800 -1820.

On the left are 'll" on both post and brace. On the right the marks on both post and brace are "llll".

 
 

 When our house was built  in 1810, the corn crib may have been here.

It has been moved at least 3 times.  It has been repaired, reinforced; its interior reconfigured. It has held corn, feed, tools, chickens, furniture, and been home to many birds.

Its plan:

A rectangle divided into 3 equal bays. The lengthwise framing in the larger space labeled 'corn' may have been flexible partitions. The tool and grain storage space has a low ceiling to allow more head room in the loft above it.

 

 

 

A cross section of the crib at the partition 

It is about 15 ft. wide at the base. Today its sill is about 24" above grade on the south end, 12" on the north.

 

 

 

 

Of course I was curious about the geometry. 

A corn crib was useful, and utilitarian. It served a purpose without frills. 

What was the simple, direct way to lay out its frame?

 

The floor plan: a rectangle with 4 corner posts and 4 evenly spaced intermediate posts. 

The dashed line shows the slope of the walls. The slope keeps rain off the corn. It's possible that the angle also keeps the cobs from locking in place. 

The slope was determined by a simple choice - see below*.

 

Today a carpenter would lay out the plan with a tape measure and a steel square. In 1810, these tools did not exist. Instead the carpenter used geometry.  

Here is the basic layout for a square as drawn by Audel in 1923.  This could have been used for the corn crib.

The layout using the geometry of a circle:

A-A is the width. Its arcs become the radius of the circle. They cross at the center of the circle. Then the circle is drawn with marks where 6 arcs cross the circle. 2 of the arcs cross at B and B: 2 at C and C.

 Draw 2 lines:  A-B and A-B. They are the sides of the  plan. 

2 intermediate posts are located at C and C, where the circle's arcs cross the lines A-B.

 Swing 2 more arcs from the corners of the plan: center of the arc on A, Then swing from the other A to C.  They cross the earlier arcs at D and D. The Line D, extended divides A-C in half and is the location of the next set of posts.

The arcs with their centers on both  D's, swung from the other D, locates E which marks the length of the  corn crib floor. (I drew only one.)

I can draw this layout much faster than I can write about it. Even so, it takes too long. I think a master carpenter would already know the geometry which I've laid out here and would have used short cuts.

 

He would have had a framing floor, probably in a barn.  

He would know the general plan. He and his client would have reviewed size and any needed variations. The work just  needed to be laid out. He would have used twine and awls, a chalk line to mark the lines on his framing floor. 

I  have drawn the chalk line set to both points, and a squiggle where the line is not held tight

A= the width of the corn crib, the ends of the rectangle.

Both sides need to be perpendicular to A: the 3/4/5 triangle  - B - sets the corner. The carpenter could then snap his Line - D - on both sides. Folding his Line -A- in half he could lay out 3 lengths along his Line, and then connect the 2 sides. The plan for the crib was done.

 

The plan could be easily 'trued' by checking that the diagonals of the whole rectangle, solid lines - E, as well as those for the 3 smaller rectangles, dashed lines - F, matched.

 

 

 

The elevation, here laid out with a compass.  

 

 

 

 Again, I think it would be easier to lay out the frame with a chalk line, using the width of the crib as the unit of the square.

 

The square and its diagonals.

Note that the diagonals cross at the location of the  beam that carries the loft. This is the first point.

 

 

2 points are necessary to draw a Line - or to snap a chalk line. The carpenter divided his Line in half, locating the vertical center line of the square.

The diagonals on the right half cross to give the carpenter his 2 points. He could locate the beam carrying the loft over the feed bins.

 

 

 

 

 

Next the carpenter needed to set the wall height. He added one diagonal on the left side; then the diagonals for the upper half of the square.

He had 2 more points whose Line (dashed here) located the plates for the side walls and the roof.

 

 

Finally he used half his unit (the width of the crib between walls) as the height to the ridge of his roof.  Here I have drawn it as an arc.

 

 

 

 

 

 * Now he was ready to layout the slope  for the walls.

A straight line (with arrows) from the outside of the sill at the bottom is on the inside of the plate at the top of the wall. This determines the slant of the walls.

I didn't discover this myself. Someone showed me. Unfortunately I don't remember who and can't give him credit. 

 

 

 

The 'front' of the corn crib today. We added a window, matching  one already in the crib.

\

The 'back' of the crib.

We moved the door to the 'back'  as we no longer unload from a wagon.

The hollyhocks self-seeded here. They approve of our work.

 

 

Post Script:  Images of the frame

 

 

Post, plates, and rafters in a corner of the crib

 

 

 

 

 3"x4" braces cut by a sash saw, pegged.

 

 

 

 

The original rafters are 3" x 4", cut by a sash saw.

Later logs were added between the rafters and butted together. Many of the logs are birch.

 

 

The rafters were mortised and tenoned at the ridge.The large peg is visible here.

Knob and tube wiring and electrical cable are still in place though today the crib has no electricity. 

At one point there was also telephone line.

Stratford Hall, Part II: The Geometry of the Elevations

Stratford Hall, begun in 1736 - As visitors would have seen it when they came to the house from the plantation's docks on the Potomac River.

 

My first post on Stratford Hall* looked at the how the foundation plan was composed of 5 equal squares. The location of the central hall which runs between the chimneys was also shown to be sited by the circle that laid out the squares.

The Master Builder, William Walker,** seems to have used  the square as his geometry to layout the house.  Did he use the square for the rest of the plans and elevations?  I obviously think he did, but we have few notes, only his actual construction as proof.

 

Was the width of the chimney bases determined by those squares?

Probably. The exterior dimensions of the house foundation and the chimney bases would have been set at the same time. Here are the Lines which divide the width of the square into 4 parts: one part on each side for doorways, 2 parts in the middle for chimney masonry.  The Lines are not quite accurate: note the red lines with arrows. Perhaps this is because the masons who built fireplaces had different skills than those who laid brick walls and the dimensions were not as critical. 

 

 

 Note that these Lines, when I added them to the square based on the inside of the brick walls, give no useful information.

 

 

A brick wall can be measured from the outside or the inside.

 A garden wall needs 2 parallel Lines (twine) to be staked - one on each side of the wall - to guide the construction.

 A brick wall for a building would also have 2 Lines. The bricklayer, checking his Lines for a building layout must decide which side of the wall to use.

 

 

 

At Stratford Hall a square and its lines define the exterior dimensions; they sets the size of the wing from ground to ridge pole at the top of the square (A).  The third points (B) mark the height of the brick walls which is the location for the  plate for the roof trusses. 

The lines which divide the square into 4 equal section locate the width of the chimney stack - note where they cross (C) in the middle of the square.

 

 

 

 

The lines of the square, based on the width from the interior side of the brick walls, cross at the floor-to-floor height of the lower level (D). This dimension tells the mason where the joist pockets for the floor joists to carry the main floor should be located. 

The central space of the house is the Hall, the place to gather and entertain. See my previous post for floor plans.*

Here the square is based on the inside dimension: brick wall to brick wall, just as it was for the wings. 

The geometry of the wings (the drawing above) applies here too: the location of the main floor and the height of the walls is the same.

The ceiling of the Hall is vaulted: a tray ceiling, sloped on all 4 sides. The height of the ceiling is not arbitrary but is determined by the height of the walls. 

 A red line encloses the hall and its ceiling;

The scale on the right side shows the wall height divided into 4 parts. The height of the sloped part of the tray ceiling is 1 part.   

4 parts = wall height, 5 parts = floor to ceiling.  

The Master Builder could have used the floor of the hall as a framing floor for this ceiling, making the height simple to calculate. 

At this small size the diagram of the square and its lines gets messy. So I chose a different way to show the relationships. 

 

* My first post on Stratford Hall:  https://www.jgrarchitect.com/2023/06/stratford-hall-and-paul-buchanan.html

** For a biography of William Walker see https://chipstone.org/article.php/559/American-Furniture-2006. He was a Scotsman, probably trained as a joiner and wright in Scotland, who immigrated to Virginia before 1730. He would have known of the work of James Gibbs and Colen Campbell. He might have known them.