Sunday, December 29, 2019

Practical Geometry - Drawing a Square with a Compass, Part 1

Draw a square with a compass? !
Here are 2 ways. There are several more.

Compasses make circles. Straight edges make straight lines. Together they can lay out whatever you can imagine.
How to Draw a Square with a Compass #1

1)   Choose a length: A-B.  It is also the radius: dashed black line A-B,  for drawing a circle with a compass.
2)   Draw the circle.

I have drawn these diagrams on graph paper, a reference to help show how the square grows.

3)  Switch ends. Hold the compass on B. Swing the arc from one side of the circle to the other: G-A-C.
Hold the compass on C. Swing the arc to find D.
Use D to find E; E to find F. along the circumference of the circle.

The circumference of every circle will always be divided into 6 equal parts by the radius of that circle. The length between each 2 points around the circumference will always equal the radius.

It's easy to draw a daisy wheel

However, to construct a square the petals are not needed, only the 6 points on the circumference.

4 )  F-G is the line. It is the same length as the one chosen at the beginning, just in a different location.

G and C are 2 points. that can be connected by a line.
So are F and D. 
They are the same distance apart so they are parallel.

A square has 4 equal sides.  (Just a reminder)
5)  An arc the length of  F-G swung from either F or G will mark  G-H and F-I the same length as F-G.  This is the same length as the chosen line A


A square drawn using Practical Geometry, using a compass.  
To check: lay out the diagonals. If their lengths are equal the square is true.   

This upstate NY barn was dismantled for reuse by Green Mountain Timber. It had a daisy wheel scribed on one wall.  The  barn laid out using the 6 points of the circle. The frame of the east elevation is drawn below.

The square frame for the door is in the center. Either side completes the rectangle of the circle.

My post describing this barn:

How to Draw a Square  with a Compass, #2

Draw a line.  Mark 2 points on the line.
Open the compass wider than the distance between the  points. Swing an arc across the line, below and above it from each point.
The arcs will cross at 2 points. Draw a line between those points. The new line will be perpendicular to the first line.

Then choose the length of the side of the square A-B. Mark it off on both lines.See the arc B-B.
Swing new arcs the same length (A-B)  from both B's.  See the dashed and dash/dotted lines. They cross at both A's.
All the sides are equal: a square.

St. Jerome's Catholic Church, East Dorset, VT, 1873, was laid out using that simple square  - including how the arcs cross each other. 

My post about it is here:

Part 2 is here:

I explain these ways of using a compass,a straight edge, and a marker to lay out squares and rectangles when I give presentations. I add them here because such information should be readily available on line.

Friday, December 27, 2019

English Construction Tools, 1669

John Leeke translated Giacomo Barozzi da Vignola's Regola delli cinque ordini from Italian into English in 1669.

While Leeke himself is worth attention; he was a mathematician, a professor, and land surveyor and well as fluent in Italian. He also helped rebuild London after its disastrous fire in 1666.  Here I want to focus on the frontispiece for his translation.

Leeke calls the translation
He adds Vignola's title.

Then he writes
For the USE and BENEFIT of
Free Masons, Carpenters, Joyners, Carvers, Painters,    Bricklayers Plaisterers:
In General
For all Ingenious Persons that are concerned in the Famous Art of BUILDING 

This cartouche fills half the frontispiece. It celebrates the tools used by all those craftsmen.
Sitting on a column base are:
A compass
A carpenter square
A horizontal level
A vertical level
A measure with a curved side and regular marks
Draped around the cartouche is a Line, its spool on one end, its plumb bob on the other.

These are the tools for people who built.

130 years earlier most of the same tools are on Serlio's frontispiece which he engraved about 1540, for his book, On Architecture.
The original is black and white. This cartouche is on the cover of the 1998 translation by Hart and Hicks.
I use the color image because it makes the tools easier to see:

A compass in front of a ring - a perfect circle.
The compass pierces the scroll of the cartouche and is held in place by the ring.
A large carpenter square spearing a tetrahedron
A straight edge spearing a cube incised with diagonals, graduated circles and squares. The size of each is determined by those on either side.
A round rule pierces both the tetrahedron and the cube.
A Line with a handle is in the lower left corner, entangled in the scrolling, ending with a knot and a tassel beside the handle.
Serlio does not include any levels.

 Walther Hermann Ryff, of Nuremberg, c. 1500- 1548, considered Serlio his mentor. He did not train under him.
Ryff was probably a pharmacist and published many works on medicine.

He also published Vitruvius first in Latin and then in German 1548.
This book, with a title of over 30 words, is referred to as Architecture. Part of the title says it is "The hardest, most necessary, belonging to the whole architecture of mathematical and mechanical art, regular report, and vastly clear, understandable information..."
This engraving is the frontispiece for his book. Many tools!
4 compasses are easily seen. 
Here's a close up of one, with an impressive arm that would have guaranteed accuracy - along with a knife, a plane, and calipers. 

Edward Shaw published his pattern book, The Modern Architect, in 1854, in Boston. It also had a frontispiece displaying tools,.
His engraving is in black and white: The color rendering is for clarity.

He includes:
A small compass in hand
A rule in hand
A large compass in the foreground
A hammer.
A large carpenter square
A cylinder which might be a chalk line
A saw in hand
A drawing and a portfolio of drawings
against the tool box:
A straight edge or rule
A vertical level
A brace is in the tool box.

A part of the original engraving showing most of the tools.

 The 12 presentations and workshops I gave this past year began with the portraits of master builders holding their compasses, and these engravings of their tools.
When my audience understood what tools were available before the Industrial Revolution they enjoyed seeing how a compass and a straight edge could layout a  rectangle, and then a building. Next they drew them themselves. They saw for themselves how the compass creates the first dimension and determines the next dimensions.
Practical Geometry was the tool  which translated a design from an idea to construction.  It was the Practical, not Theoretical, use of Geometry.


Note: our contemporary John Leeke is the 8th great-grandson of the man mentioned above. Yes, I asked and he told me so.

Saturday, December 21, 2019

The Beatty-Cramer House and Its Daisy Wheel, Part 2 of 2


Please see Part 1 for the history of the Beatty-Cramer House in Maryland now owned by the Frederick County Landmark Association.
The Beatty house was built in the 1730's. The Cramers encased that house with a new one about 1850. These posts focus on the first house, the Beatty House, which is inside the house in the photograph.
The first post:

This daisy wheel and the 3 petals - the 'eye' - were found on an original stud on the 2nd floor frame of the Beatty House.

I was asked to explore how the diagrams could have been used to lay out the frame. This is part 2 of my report.

The Beatty Cramer daisy wheel with its 3 petals, its 'eye' in red,
showing how they are part of each other.

An aside:
This daisy wheel was found on an upstate NY barn built about 60 years later. Note that it is not perfect, as the one I drew is not perfect. 
The diagram in the Beatty Cramer House is just that - information for the builder. 
The diagram on the barn sheathing was also information.*
Neither daisy wheel needs to be perfect to be useful. It's a 'setting out' tool, a record, notes to the builders and those who will work on the house later.

My first blog post laid out how the master builder would have used the 3 petal diagram to lay out the width and length of the house, the size of the 2 rooms up and down, and the location of the bents. 
A bent is a part of a timber frame , the posts and beam, that runs front to back.
This photograph shows 4 bents about 4 feet apart in the east room, first floor. The beams hold up the second floor . The posts extend above the floor of the upper room to the plate . The second floor on the east side of the house has half walls.

The second photograph shows how the west side of the house was originally stepped down. See the tie beam which is being photographed in the center of the existing wall on the left. That beam was part of the floor frame for the west side upper room.

The drawing shows the east and the west end elevations.  The floor and wall frames that were part of the Beatty House can be determined by the mortises left behind when the frame was changed. The pitch and frame of the roof, especially the east end, is not as easy to ascertain.

The master carpenter has his first dimension: the width of the house, and he has left us his note about how to apply that dimension: the  red line at the right is the width of the house. The arcs of the eye is the way he sets out a square, finds his next dimension. 


A note about laying out the daisy wheel to find the next dimension: the carpenter could have used a Line to draw the arcs. He could also - when he wants square -  have used a long compass like then one shown to step off the dimensions. Every rectangle could be trued by checking the  diagonals, as framers do today.   

The framing elevations
First the west end - the frame on the right in the drawing above.

The width of the house becomes the radius for the arcs that determines  the height of the bents. The tie beams and the plates are set where the arcs cross This is the same geometry that determined the floor plan of the house: the 3 petals of the daisy wheel, the eye - see the previous post.

The height is divided in half to determine where the 2nd floor will be located.

The plates extend from this west side of the house to the east end. The builder has to work with that height. See the red arrow from the west to the east elevation. 
The east rooms are a few feet higher than those on the west.  How did the builder set that floor height?
He may have used the height where the diagonals cross the arcs to place the 2nd floor of the east side. See the black dashed line. The daisy wheel could then have set the sill location.

The edge of hearth in the 1st floor west room also uses those points.

Or perhaps the difference in height was set by daisy wheel from the beginning.

The house was built into the slope of the land. The simple way to excavate the storage cellar under the east rooms could be by digging into the hill from the west end. The kitchen on the west  needed to be only a few feet above grade - for easy access to the spring house. The formal chamber on the west also needed to be a few feet above the grade to keep the wood sills dry. 

I have indicated the slope of the land with a black dashed line below the framing drawing of the north elevation.  The grade on the west (left) side is as seen today.  The grade on the east  (right) side would have been at least 12" below the top of the stone foundation.

The dirt removed to create the cellar hole could have been been used to level the area on either end.

The diameter for the daisy wheel for the west rooms is the height from the sill to the plate. 
The daisy wheel is drawn with its axis horizontal to show the location of the second floor joists.
The diameter for the daisy wheel for the east rooms begins at the sill and ends above the plate. The petals of the daisy wheel are located at the underside of the plate. Here I have drawn the daisy wheel with its vertical axis. The builder would have begun his layout from the under side of the plate, which was a given.

The daisy wheel on a horizontal axis could locate the plate for the east 2nd floor ceiling. The braces could have extended to that height.  The petals already locate the studs on each side of the fireplace.

The roof ?
It's not there. Nor is the ceiling height for the east 2nd floor chamber. Only an original window location, the stud locations, and a brace with an angled cut from a reused collar beam remain. I have extended it here.

2 roof pitches have been proposed.

One, shown here, matches the Jan Breese roof built c. 1723, in New York State.
That roof would be based on the daisy wheel and a second interlocking circle. It is not a diagram used for other parts of this  frame.

The roof pitch could also be based on the eye - or as described: rafter length 3/4 the length of the roof span. The arc of the width of the  house fits the lower pitch. 
The arcs have arrows.  I have added the roof line.The plates are red.

I prefer this pitch because it comes from the geometric notes the builder has left for us. It is consistent with his overall layout.

It is time to look again, on site, with twine to mark the dimensions, swing the arcs,  and especially, consult  directly with the members of the Frederick County Landmark Association. 

*How that daisy wheel was used is in my post:

Friday, December 20, 2019

Beatty-Cramer House and its daisy wheel, Part 1 of 2

An Introduction 

This house in Maryland was about to be burned down because its owner had no use for it. It looked to be a c.1850 farm house that might have also been an inn.

A local historian asked to look at it. 
He had seen the frame and the brick nogging (infill between the posts used for insulation) where the siding was missing on the end. Both looked to be Dutch, not English; the spacing appropriate for a Dutch jambless fireplace. The post below the window had been cut to make room for the window.
What had it been when it was built?

Inside he found another house, 1-1/2  stories: the main room on the north end with beams 4 ft apart supporting the 2nd floor.
The original kitchen to the south end had been down half a story. It opened out on grade to the well house  a little farther down the slope.

This 'inner house' is the Beatty House, built in the early 1730's by the family of Susannah Beatty who moved here from Ulster County, NY.  The property is now owned by Frederick County Landmarks Foundation.
The Beatty House is 20 ft. x-.40ft., story and a half with a lower kitchen, and an H-bent timber frame, an Anglo-Dutch style found most often in New York. 
The well house, just south of the main house, had a well below and a smoke house above. The Foundation has restored it.

In the 1850's, new owners, the  Cramers, dramatically remodeled. 
The windows were relocated; the old openings blocked. 
The half story split of the wings was eliminated; the 2nd floor gained full head room. 

All of this was done by building a new structure around the original  house. 

The photographs shows the original 2nd floor half story with the new house wall behind it. The views are to the northeast corner and the southeast corner. The end of the plate - where it was cut off  for a window - is visible in the northwest view. I've added a close up view below.

The southeast view shows the new windows cut into the original frame. Also note the original nogging in all the photographs. 



On the second floor the plate, which had been carved to be the fascia and soffit for the roof overhang, was left in place. The new house was simply framed outside of it. 

As shown above, much of the history of the house is visible in the frame. Here is the mortise for a missing beam.

There are remainders like this all over the house. The original frame can be read, and the later renovations too. The attic over the raised south end - the c. 1850 kitchen and bedroom wing - was framed with reused rafters: many with scribe marks but not in order.  

The Geometry

These 2 diagrams are on  a stud on the second floor.
The daisy wheel is about 3.5 inches across.
The 'eye' - which is  1/3 or 3 petals of the daisy wheel is about the same size.

Could I discover how/if these diagrams governed the layout?

The Foundation had measured drawings of the Beatty House - the original house. They gave me a set.
I began to look.
Of course the builder considered the land, the size of the house, what rooms were needed, how situated. He knew how he would frame, where the windows and hearth were to be located. Much of this would have been tradition.
The siting was specific to this location: the wall facing the road and the width of the house were known before the actual staking of the Lines.

This is framing layout for the bents to support the second floor.
East is to the right.

The east room is a square. The west room is less than a square. The framing is shown for the jambless fireplace in the south room, that for the north room is missing. The ends of the tie beams are still there in the exterior east wall.

I found that the plan could be determined by the 'eye', the 3 petals of a daisy wheel.
Here is an introduction to how the diagrams can be used.

 The first length is the Line A-B, the width of the house.
The Line becomes a radius for the arc  B-B-B.

Switching ends of the Line - using B-A  instead of A-B as a radius - creates the arc A-A-A.

 Both arcs can easily be laid out together. They create the Eye: A-C-B-D.

Using B-A as a radius  a circle is easy to draw - A-D-A-C-A .

6 evenly spaced points on the circumference of the circle are easy to mark:  A-D=G-F-E-C-A.

I added the arcs - the daisy wheel petals - to reference the daisy wheel that was found on the stud.

Every circle is governed by its radius.
The radius creates the circle. The circumference is divided into 6 equal parts using the radius. 
The dimensions for the radius and the segments of the circumference always are the same. 
A-B = A-C = C-E = E-F = F-G = G-D = D-A .
A-F = C-G = E-D.  

Don't groan! 
There are people in my workshops who need this step by step instruction.
There are people who tell me this is poppy cock. 

Both groups have little knowledge of geometry, and thus no understanding of  how Geometry can be Practical or why a compass is a construction tool.

To reiterate: The eye is 3 petals of the daisy wheel. The length of one petal is the distance between the petal tips. It gives the builder his first dimension from which all the others come.
Here it is the width of the house.  Thus C-E is the  same length as A-B.
B, the center of the circle, is only a dot here for clarity.

C-E, C-D, and E-G  can all be drawn - all are straight lines between 2 points. Adding the center line A=F determines the rectangle that is the west side of the house.


Extending Lines C-D and E-G beyond the edge of the circle and then using the radius to lay out a square for the east side of the house completes the floor plan. 
Note that no measurements were needed to lay out this plan. All of it was done using the length of the width of the house. Twine and pegs  - and a good mind - would have been tools enough.



I first thought the house was laid out as described below.
Today I think not.
Still this is a clear clear description of how a house could be laid out with a given dimension, a Line (twine), and pegs. 

The 2 squares can be seen as 2 half circles. 
The front wall is A-B;  initially it is a Line of unknown length.  The width of the house (C-D) is the radius. It is set where the builder thinks the wall between the 2 rooms should be.  With C as the center of the circle swing an arc from A to B. 
Then with D as the center of a new circle, swing an arc from E to F.
G marks the intersection of both arcs.
The line perpendicular to A-B and D-C-E  at G, parallel to C-D  is the inside edge of the south wall of the south room.

G is located on the inside of the west wall. I've seen many barns and buildings whose dimensions begin from that side, or perhaps they end there.

Many buildings were laid out from the 'in' side of the sills. See my post of the Streetsboro, Ohio, Baptist Church.

The bents for both rooms are easily laid out by the Rule of Thirds, begun at the inside of the sills. See the red rectangles laid out on the inside of the sills.
The narrower west room placed the bents symmetrically: 2 on the west sides, 2 on the east sides of the Lines.
The east room placed the bents on the east side of the Lines. This gave a little extra room for the jambless fireplace.
See the red spots on the sides of the bent.
The east and west sides of the house are not the same width. But once the spacing for the fireplace bent was set, a compass set at the right width could have been used to mark the other bent locations. Or the Lines could have been laid out on a framing floor.

The supporting side beams for the jambless fireplace were located at the quarter points of the exterior wall.
The builder could have marked the inside width of the west wall on his Line, folded the Line in half and then in half again. He would have the quarter points. 

The elevations and the daisy wheel are in 'Beatty Cramer House and its Daisy Wheel, Part 2 of 2'.

For an introduction to the Rule of Thirds see: