Bibliography for Practical Geometry, aka Architectural Geometry

A list of books by and about builders and architects who used Practical Geometry; many are primary sources.
At the end are other sources, websites, and credits.
It does not include books I refer to in a specific post. For example: Audel's Carpenters and Builders Guide, Theo Audel & Co. Publisher, NYC, 1923, is footnoted at the end of that post: https://www.jgrarchitect.com/2018/11/lines-in-historic-and-modern.html.

A few are books on architectural history and technology which I reference regularly:  Bannister Fletcher's History and Knight's American Mechanical Dictionary.

PRACTICAL GEOMETRY   aka ARCHITECTURAL GEOMETRY  

  

BIBLIOGRAPHY 

Benjamin, Asher. The Country Builder’s Assistant, 1797, Dickman, printer, Greenfield, MA –

     reprint by Applewood Books, Bedford, MA.

                     *The American Builder’s Companion, 6^th edition, RP &C Williams, Boston, 1827

 

Biddle, Owen. *Young Carpenter’s Assistant, published by Benjamin Johnson, Philadelphia,

     1805.

Charles, FWB. The Great Barn of Bredon, Its Fire and Reconstruction, Oxbow Monograph 76,

     1997, Oxford Books, Oxford, UK.

Colvin, H.M., A Biographical Dictionary of British Architects,1600-1840, Harvard U. Press, Cambridge, 1954.

Fletcher, Bannister. A History of Architecture on the Comparative Method, Charles Scribner’s

      Sons, NY, 17^th Ed. 1967.

Gibbs, James. *Book on Architecture, London, 1728

                       Rules for Drawing the Several Parts of Architecture, printed by W. Bower for the

      author, London, 1732, ECCO print edition

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

Green, Bryan Clark. In Jefferson’s Shadow, the Architecture of Thomas R. Blackburn, Princeton

       University Press, NY, 2006

Harris, Eileen. British Architectural Books and Writers, 1556-1785, Cambridge U. Press, Cambridge, 1990

Harris, Leslie. Robert Adam and Kedleston, The National Trust, London, 1987.

Knight, Edward H. American Mechanical Dictionary, Vol I, II, III; J.B. Ford & Co. NY, 1874.

Macaulay, David. Mosque, 2003, Houghton Mifflin co., Boston. His many books are excellent references. 

Nicholson, Peter. The Carpenter’s New Guide, 1793, London; 10^th ed., Philadelphia, 1830.

Palladio, Andreas. *The 4 Books of Architecture, 1570, translated and published by Isaac Ware,

      London, 1738.

Serlio, Sebastian. On Architecture, Lyon, France 1530, translated in1611, available on-line. 

       Translated by Vaughan Hart and Peter Hicks, 1996, Yale University Press, New Haven

Shaw, Edward. *The Modern Architect, Dayton & Wentworth, Boston, 1854

Smith, Laurie, The Geometrical Design of St. David’s Cathedral Nave Ceiling, A Geometer’s

      Perspective, The Geometrical Design Works, 2017, printed Exeter, UK. and others.

Vitruvius, Marcus. *The Ten Books on Architecture, c. 10 BCE, translated by Morris Hicky

      Morgan, Harvard University Press, 1914.

Ware, William R.. The American Vignola, * of the first edition published by WW Norton& Co.  NY, 1977 ( original edition, 1903)

*Reprinted by Dover Publications, Inc., Mineola, NY

Drawings:

     HABS drawings, Library of Congress, Washington, DC

     Denison Bingham Hull, Old First Church, Bennington, Vermont, c. 1935.

     James Platteter, barn frame for Green Mountain Timber Frames, 2014

     All others: Jane Griswold Radocchia

 

Web sites:

     www.jgrarchitect.com (you are here!)  and  www.janegriswoldradocchia.com

for Laurie Smith: 

     http://historicbuildinggeometry.uk/ and http://www.thegeometricaldesignworks.com/   

 

See also : Instagram   'jrarchitect'

 

 

This blog post is subject to updating.  

It was first complied to accompany a lecture on Practical Geometry Geometry in 2016. It is now available whenever I present.

 

Geometry of the Old First Church Fanlight - an Addendum

The first post about the design of the fanlight is here: https://www.jgrarchitect.com/2012/12/old-first-church-bennington-part-2.html

Considering the 'leaves' of the fanlight,  those 3 curved petals that fan out from the base of the light: how did Lavius Fillmore, the master builder, and his crew, especially Asa Hyde, the joiner, derive the pattern?

The layout that I drew of the leaves in the last post is too complex.

 

 The church is graceful and direct. 

 

The derivation in the previous post of the leaf pattern is not direct.
The geometry of the church is straight forward. The use of the circle to layout the framing, the design, would have been clear to other people in construction and to the church's congregation, as well as to anyone in that time who was educated beyond grammar school.

      

 

 

And then there's this diagram in my last post: 

I drew the way the scallops on the curve of the light can overlap simply by rotating the circles one half a petal's width around the circumference, or 15*.

It is also the pattern of the leaves, just at a scale too big for the fanlight.

 

 

 

Drawn smaller, the pattern has 3 overlapping circles at its center, across what would be the sill of the fanlight.  Here the circles come first; the leaves come from the pattern; the fanlight, its size and placement, come from the width of those combined circles.   

However, the pattern in the The Old First Church fanlight was laid out knowing the dimensions for the fanlight. The door and its surround, the placement and size of the door in the main elevation, the width and height of the fanlight were determined by the geometry of the building. They were fixed.

So: given the width and height (about 60"w x 30"h)  how were the leaves' sizes determined? 

The 3 circles across the sill were overlapped. If they were 3 in a row the proportions would be 1/1/1. Then the width could be divided into 3 equal parts. Instead the proportions of the circles are 8/6/8, or 22 equal parts.  Dividing a line into 22 segments with a compass and straight edge is complex.

 

The center lines  (faint pencil lines here) of one set of the fanlight's scallops meet at the center of the fanlight  These are the scallops that the leaves point to.

The distance between the scallops and the center of the fanlight is the diameter of the circles that will make the leaves. The first circle crosses the pencil lines and marks the center of the next circles.

The red spots are the centers of the circles.

 

  

The first 3 inner circles. Where they cross each other and the sill they mark the centers of the next circles.

 

The 4 outer circles. The ones that continue below the sill are not completely drawn.  Note that even though the center circle begins the design it was not needed here. It was understood implicitly by the joiner laying out the pattern.

Below is the layout of the leaves with all the circles included. 

 

 

 

Geometry of the Old First Church Fanlight

 

 

This is the fanlight over the main door to the Old First Church, built in 1803-5 in Bennington, Vermont. Lavius Fillmore was the Master Builder; Oliver Abel, his Master Carpenter, and Asa Hyde, the Joiner and carver.  

 The fanlight design consists of 2 parts: the 'scallops' around the curve and the 'leaves' coming up from the base. It is simple, graceful.

How was it laid out? In 2012 - when I first wrote about this fanlight - I knew the geometry for the scallops around the curve - expanded daisy wheels on the horizontal and the vertical axis. 

The 3 leaves below the scallops?  I was lost.

Laurie Smith - English timber framer, historian, geometer, the most knowledgeable person I know about the use of circle geometry in medieval design and construction - provided an answer.
Here was our geometry for the fan light in 2012.  Now, in 2021, I think it is probably not how Lavius Fillmore laid out the pattern. 

I post it because we are teaching ourselves. We are constantly learning more about how to use Practical Geometry (or 'Architectural Geometry'). One solution is not the only one. 

A square can be  derived from a circle in many ways - as can a 3/4/5 rectangle - both done using only a compass, straightedge and a marking tool. To see how the geometry can be followed and bring different designers seeing different paths, arriving at the same solution, is valuable.

 

 

 The  circle, its 6 points around the circumference laid out by the radius of the circle, is set on a line which defines the shape of the fan light.

 

 

The circle is surrounded by 6 circles which have their centers on the 6  points. The center pattern is a daisy wheel with 'petals'.

 The circles expanded.

 

 

This set of circles around the original circle adds petals 
to the exterior of the first circle. Add the fanlight shape and the petals  become scallops around the arc of the fanlight.

 

 

 

 

 Rotate the circles 15*  - or 1/2 a petal - and the fanlight's scallops' locations change.

Overlapped, the daisy petals create the double scallops around the Old First Church fanlight. See the photograph above.

 

 

The overlapped petals are also the pattern of the 'leaves' in the  fanlight: too big, not in the right location, but crossed as are the leaves. 

This pattern is much easier to layout than the geometry in the next 5 drawings. I include those drawings to show another way which is equally valid but more complex. 

However, as I have explored the Old First Church geometry over the past 10 years,  I have gained a sense of how Lavius Fillmore designed. I do not think he used the sequences laid out here. He understood what the simple act of turning an expanded circle on its axis could create. 

The simple solution is here :

https://www.jgrarchitect.com/2021/10/geometry-of-old-first-church-fanlight.html 

 

 

Here is how the leaves could have been added. These steps are our 2012 solution.

Add regulating lines from the center of the circles to the second ring of circles and center lines in the petals.

 

Connect the center points of the scallops to each other. Where they cross the petals is the center of the small circles which form the leaves. 

 

 

 

 

 

 

The radius of the circles is the distance from the center of the petal to the scallop.

 

 

 

 

 

 

 

 

I've drawn it in red to make it more visible. It is a complex layout for a seemly quiet, unassuming design.

This pattern was drawn at about 3/8" = 1'-0".  A scale of 1"=1'0" might have been easier. However it would still be tiny here on the page. For clarity I left out the overlapping scallops.

 

 

my drawing in 2012:

The real fanlight was laid out full scale - 5+ feet across -  on a framing table or floor. The proposed design sketch would have been studied, the arcs drawn with a compass using chalk or charcoal, the lines checked, redrawn, the points pinned.  Finally, when the regulating lines were erased, the simple, clean design was visible.

I would like to have been there, listening, watching. as the men drew this.  I think they were pleased as they derived the pattern and settled on a design.  It's not structural at all. It's one of the first things you see, an introduction to the church. It's also their signature.

Then -  I realized that this derivation is not simple enough. 

My Addendum, my next post, is my current solution, my current understanding of how Lavius Fillmore, Oliver Abel, and Asa Hyde designed the fanlight. 

https://www.jgrarchitect.com/2021/10/geometry-of-old-first-church-fanlight.html

Rockingham Meeting House, Rockingham, VT

The Rockingham Meetinghouse  was begun in 1787, dedicated in 1798.

 After some preliminary analysis of the design and frame I realized in 2014 I needed to see it. I wondered if it would be as spare as the Rocky Hill and Sandown meetinghouses which preceded it.

It is.

 

 

 The site, on top of a hill with a view all around, emphasizes the simplicity of the structure. One can only come to it from below, and like many 18th c. New England buildings it sits upright and confident. It is very impressive.

I returned in 2018 and early 2020,  I updated the geometry as I learned more. I revised the drawings again when I gave a Zoom presentation in spring, 2021. What I first saw as a complicated geometry became simple and direct. 

This is not a dramatic design created by a London architect like Robert Adam to wow his rich patrons. It is a meeting house for a rural community. It is a straightforward layout planned by master builder, John Fuller, for a simple timber frame to be erected by a crew of local citizens.


The Town Fathers specified a building 44 ft. by 56 ft. The HABS drawings read 44'-4" x 56'-6". The difference could easily be the addition of the sheathing and siding to the frame. The porches (the end staircases) are square: 12'-2" x 12'-2". 
The difference could also be that the rule used then and the one we use today differ slightly. I am not sure they had a 'rule'. Poles of various lengths, 4ft., 5 ft., 10 ft., are in some illustrations.

 

'General' John Fuller, the master builder, was also the architect, engineer, framer. He knew the meeting house required an open space in the middle so everyone on the floor and in the balcony could see the preacher in the pulpit - and be seen by him. The pulpit was centered high on one wall, a window behind,the balcony on 3 sides.

 

He laid out a 3/4/5 rectangle. noted here in red. Then he laid out a square in the middle which defined the open space and divided that into thirds to set the columns for the balcony and the posts for the frame. See the black square and columns.


He extended the column spacing - the dashed black lines - to place the posts on the front and rear walls.This made the balcony the same depth all the around.
The porches are squares set in the middle of the west and east walls. The exterior posts were set at the porch corners, not at the 1/3 points of the wall. This also allowed for 2 windows on each side of the porches. 

 

 

 

Those 4 closely spaced posts in the center support the 4 attic trusses which are braced together to span the width of the church and allow the center of the meetinghouse to be an unobstructed space.

Walter Wallace, standing in the joined trusses under the ridge, gives a sense of how big the framing is.

 

 

The HABS prints of the End Elevation and the Interior Section for the Rockingham Meeting House are hard to read but their basic dimensions are clear.

The End Elevation is composed of 2 squares. The roof is framed using the 3/4/5 triangle. 

The notation to the right of the rafters says the pitch is a 9/12, modern language for the same thing.

 

 

The porches, the name for the stair towers, are set in the center of the end walls. The diagram shows that if the overall width of the wall is 8 modules (each square divided into 4 equal parts) the porch  is 2 modules wide.  

The proportions are 3-2-3, a graceful rhythm. If instead the massing had been 3-3-3 - all the widths equal - it would have felt dull. 

John Fuller, Master Builder, understood how to create with those simple shapes.  

 

 

The Interior Section shows the roof trusses, all using the 3/4/5 geometry. I've highlighted them in black for visibility

The meeting house height is divided in half horizontally. The columns which support the balcony divide the width of the meeting house in thirds.

If those columns has been the posts in the exterior walls only 1 window would have been possible on each side of the door. As I described above, the posts in the exterior walls were set differently (the black dot/dash line). 

 

 

 

 

With his frame laid out, John Fuller now needed to place the windows. The 6 posts on the front elevation were fixed. To allow any visual space* between the 2 windows on either side of the main door had to framed against the posts.  Here you can see how they were placed; there is no room for casings. 

* 'Visual space': the windows needed to be viewed as separate shapes, not as pairs.    

 

 

 

The red lines on the front elevation show the locations of the 6 bents for the meeting house.

 

One more window was needed on either side of the main entrance.

Where would they fit so that they were part of the whole, not call attention to themselves, and enhanced the main entrance ? 

Fuller used geometry to place the outer windows in relationship with the others. 

On the right side of the entrance is the front elevation as it was built.

On the left the outer windows are shown set in relationship not to the posts, but to their next closest windows and the left side of the elevation. The entrance is flanked, but not crowded by the windows.

 

 

The 'empty' wall to the left becomes part of the geometry. It shares the  proportions, being 1/4 of the wall. It is not 'left over'.

 

You can see the design succeeded. The uneven spacing between the windows is interesting and enlivens the facade, but it does not detract from the main door with its pediment. On either end the stretches of wall without a window anchor the meeting house to its site. 

 

 

Here is the main door. Its height is the determining dimension. Half the height is the radius for a circle and its square, drawn in red. The rotated square is drawn in black. The intersections determine the width of the architrave, the columns. The location of the plinth blocks and the depth of the moldings in the architrave, over the door are governed by the sides of the smaller square to the original circle.

The pediment follows Serlio's instructions:
 half of the width dropped below the base of the pediment - black lines - becomes the point for an arc whose radius -dashed red line - is the distance to the edge of the pediment. The dropped line is extended up to the arc; that marks the height of the pediment.

 

 

 

 

 

 

This way of laying out a pediment is shown in Asher Benjamin's 1797 pattern book:

 

 

The door itself came after the frame was in place. It was built to fit the opening. 

First the door's rails and stiles were laid out. 

Then the Rule of Thirds divided the remaining space in half and sized the panels and the stiles between the panels.

 

 

 

 

Last picture: The windows on the sides of the meeting house were framed against the posts as they also were on the front and rear elevations.

Just as at the Rocky Hill Meeting House in Amesbury, MA, the eaves on the porches bump into those windows. Neither master builder had solved that problem.

 

For excellent information about trusses in meeting houses and churches see Historic American Roof Trusses, Jan Lewandoski, et al., published by the Timber Framers Guild, 2006. www.tfg.org.

The Rockingham Meeting House is not included but the theory, practice, and evolution of the trusses used for similar meeting houses is laid out with clear photographs and Jack Sobon's drawings.

If you do not know how to use of the 'Rule of Thirds' square as a design tool, see: https://www.jgrarchitect.com/2020/08/lesson-6-rule-of-thirds-part-1_21.html