Tuesday, September 22, 2020

Lesson 7, How to layout a frame with Lines

 Lesson 1 is here: 

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


Friday, August 21, 2020

Lesson 6: The Rule of Thirds, Part 1


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

Thursday, August 6, 2020

Owen Biddle's 'Young Carpenter's Assistant' , Plate I, G

A note on Owen Biddle's Plate I, Diagram G. in his pattern book for beginning carpenters. *

I wrote about Diagram G on this post: https://www.jgrarchitect.com/2020/06/practical-geometry-lessons-lesson-5.html

I said that Biddle was not just introducing his 'carpenter assistant' to geometry; in Diagram G Biddle was explaining how to layout a square corner to work out a structural detail, cut a board, or set a frame on site.

Since then I have explored the theoretical geometry of that diagram.

The number of right angles which can be drawn in a circle is infinite. The rule always works. That understanding is part of why geometry is seen as mystical or sacred.

This 'squaring the circle' diagram is from
Robert Lawlor's Sacred Geometry*. (page 77, diagram 7.5)
It uses a geometry similar geometry to Biddle's diagram G: a diameter and an angle. Here the diameters are evenly spaced and the same angle  is used at every point on the circumference. But the angle is not 90*. It is not a 'square angle'.
This is decorative, not structural.
The shapes do not close. The line continues for 5 rotations. It does not create a square, but seeks to define the perimeter of a circle with straight lines. 
I am often told that I work with Sacred Geometry, that the geometric patterns I recover are theoretical, mystical, and sacred. I agree they are geometry. No, they are not sacred. They are practical. They are geometry used in construction.

Here is how Biddle's diagram comes about: 

Begin with  a point  - A

Choose a radius - A-B,  and draw a circle. Using the daisy wheel find the diameter - B- A- C, dotted and dashed line.

Pick a point on the circumference of the circle - D.

Here I have chosen 3 different D's  at random.

Connect B-D and D-C.

Each diagram will have a 90* (right) angle at the intersection of  B-D-C.

Wherever the D is placed. the angle will be 90*.

Biddle's Diagram G begins with my line B-D.
It describes how to find my 90* angle of B-D-C. (his a-b-c) The answer is to find the diameter of a circle (a-d-c) that intersects a. That will give c. That will give the 90* the carpenter needs.


By Hound and Eye* has a very similar diagram for drawing a right angle .
The book is a  guide to furniture design, full of practical geometry. Each geometric problem is described step by step; practice work sheets are included.
This pattern is the beginning of a handmade try square. 


*Owen Biddle's The Young Carpenter's Assistant, 1805, Philadelphia. Dover Publishing  reprint,  See my Bibliography for more information.

*Robert Lawlor, Sacred Geometry, Philosophy and Practice, 1982, Thames and Hudson, London.

*Geo.R. Walker & Jim Tolpin, By Hound and Eye, A Plain & Easy Guide to Designing Furniture with No Further Trouble, 2013,  Lost Art Press, Kentucky The diagram shown above is from page 57.

This pattern is 4 overlapping hexagons.
My granddaughter, who is 7, watched me add the images to this post.
She wanted us to 'square the circle'. I did, using right angles where the diameters met the circumference. That produced these overlapping 6 hexagons, not squares.

She watched closely and observed that accurate work was not easy: my lines did not always cross exactly in the center of the circle. When we finished she asked me to erase all the diameters. This is the result. Maybe she will show me later what she added to the copy I printed for her. 

Saturday, July 18, 2020

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.

Sunday, July 12, 2020

The Miller's Toll, Bennington, VT - its construction

The Miller's Toll is a restaurant in Bennington, Vermont. 

This post is about the building's construction.

The current  owners knew they had an old building when they began renovation in 2017. 
I asked to explore the place while its frame was visible. They agreed, with pleasure.
I wrote about its history on my blog, 'Passing By'*.

The main house, now surrounded by first floor wings and a second floor jut-out, is a post and beam frame with plank walls.
This framing system is not uncommon locally. Occasionally plank walls were used in western NY and Ohio; indicating that framers who had learned their trade here built there. 

First:  what I saw and some history.

This is part of the plate for the roof of the back wing. It was now part of the 2nd floor wall frame.
This back wing may be the original house - a small raised cape (half walls on the 2nd floor - modified Anglo-Dutch frame) typical for this part of Vermont.
The main house may have been added as the owners and the town grew.

The building is on every map we have of the town. It is a black spot on the first - the 1835 Hinsdill map. It is also a spot without a name on the 1856 Rice and Howard map.


This is part of the 1867 Beers Atlas map.The house is in the middle with the owner's name (illegible) jutting up. M C Morgan's house - now the Safford Inn - is just to the right, across the Walloomsac River.
The Safford family were early settlers of Bennington. They built the house and ran the corn and saw mills across the road. The mills are depicted on both maps. M.C. Morgan inherited the house.

Here is a small part of the 1877 Bird's Eye View Map of Bennington. and the same map updated in 1887.

In the middle, beside the Walloomsac River,  above and left  of the bridge, is the house.
It is a 2 story house with a back wing and a porch on 2 sides. There are 3 windows on the second floor in the front and a chimney in the middle.

It appears that the front porch was enclosed by 1887, or perhaps the delineator was more skillful.


The hole in the roof for the chimney is now patched. It is right where the map placed it.

It's the house which was here in 1877.

The front wing had one chimney and no fireplaces. Cast iron stoves were manufactured locally as early as 1820. This house probably had one. 

 A view inside the 2nd floor of the front wing, probably built   before 1830. 

The 3 windows seen on the map are there.
The roof has the same pitch with the gable facing the street.
The bunch of wood in the  photograph in the middle of the floor - where the surface changes -  covers the hole where the chimney was.

The gable of the house faces the street, an early step in the evolution from late Georgian to Greek Revival vernacular architecture. However, I could see no framing showing a stair had been located in a hall on one side of the front wing, also a hallmark of Greek Revival.
A simple stair was set between the back wing and the new one.  The pale sliced rectangle, lower left, is the first step for that stair. Its frame was not visible, its moldings cobbled together - I couldn't date it.

The ridge beam, running down the middle of the picture, has 5 sides. These ridge beams were standard in Bennington houses from c. 1770 to the Civil War. Wide boards with wane were used for roof sheathing.

The ceiling joists ran parallel with the  ridge, set into the beam. This is also common locally. In the photograph a later ceiling frame is barely visible.
I saw no scribe marks; this is a square rule frame.

The post and beam frame, painted here in the photo, is typical of New England timber frame construction found in Bennington from 1765 into the 1860's.

The walls have no studs. Instead planks sit side by side.  Bennington had lots of wood. Water powered saws quickly cut that wood into many wide panels. The intermediate studs we had earlier used in the post and beam frame gave way to plank walls.

The availability of wide boards was due to advances in saw mill technology. The contemporary sash saw could cut several boards at once; earlier saw cut only one board at a time. The proliferation of these boards may be part of why we began to add wide corner boards, wide frieze boards under the eaves, to outline pediments in gables -  to cover our simple, traditional house shapes in Greek Revival decor.  

Those planks were cut at Safford's saw mill just across the Walloomsac River.
The sash saw blade went up and down. It left the marks on the boards which are still visible today - the left side in the photograph.

On the right side are the light and dark marks left from where lath was nailed on for plaster. The uneven lines mean that the lath was 'split'  - made from  boards. Later lath is all one width cut instead of being split. 

In1896 the Sanborn Insurance map labels the house a Cigar Manufactory" . Here is the owner's advertisement in the 1896 town directory.
Later it became a market, then a restaurant - The Vermont Steak House, Peppermills, and now The Miller's Toll.


When I posted a link on a local history page a lively discussion took place on cigar manufacturing and small town employment in the early 1900's. The outer layer of the cigars came from Connecticut River Valley tobacco fields, the delivery made possible by the railroad to N. Adams that used the Hoosick Tunnel, an engineering feat for its era.
Young women with nimble fingers were employed to roll the cigars. Immigrants were usually hired as speaking English was not required.

*I wrote about it in my local blog: