The layout of an Italiante window pediment

  9/14/25: my last diagram is not quite right. I will update it.

 

The curvy Italianate molding above that window in the center?  How did they do that? 

How did carpenters in south eastern Massachusetts in c.1850 lay out the picturesque pediment so it could be cut and assembled?  

 

 

This was the question posed to me by Nathan Goodwin of H.I.S. Construction. He was asked to lay out a copy of that molding for installation above two garage doors. 

Nathan and I posted ideas back and forth. Nathan needed to draw and define the curves, especially how the arc over the shuttered windows evolves into the arc which ends in a point above the main window.

While we focused on that curve I wondered how the master builder laid out the whole design. What geometry might he have used so that the pediment complimented/completed the windows? So it is integral, not just stuck on the top?  

 

 

 

 

 

 

The first task was to see the form that was there: 

A center window flanked by 2 skinny windows, now shuttered. Together with their casings they form a rectangle. The 3 windows' tops are half circles. Around and above them is the embracing molding which follows the semi-circles before it swoops up with a reverse curve to meet at the center in a point. The molding seems to be the same width throughout.  

We saw that the pediment was derived from the windows. The windows and their casings were a rectangle, 4 units wide. Note the red  rectangle and lines below the window sill.  

 

The curve for the section of the pediment over the side windows was easy to see and draw. It's the extension of the small windows' half circles. The red dashed line and the dotted lines drawn here radiate from the center of those shuttered windows. I shared this with Nathan. 

 

The pediment's height over the center window was also easy to find - see the dashed red arc on the right. The half circle's radius is half the width of the windows.  Note the black dashed line. It's a reverse curve. Where was its center? How was it generated?

The white-out on the image comes from us exploring and rejecting options.   

Nathan and I shared ideas back and forth. He posted this suggestion: a layout based on the width divided into 4 units.  

 He extended my 4 units into rectangles, divided the rectangles themselves into 4 parts and used 3 units as the radius for the molding hoods over the shuttered windows. 

The sides of the rectangles cross the semi-circle. He added diagonals from that point to the center of the top of the main window. The crossing point became the center of several exploratory circles for the reverse arc curving to the pediment's center.

 

 

Nathan ended up with this diagram. It worked for the width of the garage doors; he could lay out the curves and cut the parts. 

 
 

 

 

 

I wanted to know about the original design - the layout of windows, casings, pediment with moldings. How might the master builder/ joiner/carver (I don't know his title) have laid out the design? 

 Here's what I saw.  Nathan's division of parts worked across the width of the window. The whole width is 16 parts/units.The center window is 6 parts wide; the casing on each side is 1 part. The side windows are 3 parts wide; with one part on each side for the casing.  

Nathan's geometry for the reverse arc over the main window also worked. The radius of the center window's arched top is 3 parts, with the casing: 4. This doubled is the diameter of the circle which draws the arc of the  pediment.

 

 

   

   

 

 

 

 

 

 

 

The Carpenter Square and the Compass - The Evolution of Practical Geometry

 

On May 31, 2025, I will present Practical Geometry and Carpenter Squares at the Early American Trades Association (EAIA)* conference in Rochester, New York. I expect I will be introducing Practical Geometry and then explore how the use of a carpenter square began to change the visual character of our architecture. I hope to see tool collections and hear other members' thoughts.

 What happened after 1820 when the carpenter square became a reliable drafting tool? When the compass, line, and scribe were joined by an L shaped piece of steel with a dependable, true 90* corner?

The squares shown here were made in southwestern Vermont c. 1830-50. They now live at the Bennington Museum, Bennington Vermont, and can be seen by appointment.

 

 

 

 

Here you can see the hand stamped numbers on the earliest squares as well as carefully drawn scales. Were the scales as important to the builder as the true 90*angle?

?

    

The square made design and layout accurate in fewer steps. Units (inches and feet) were uniform, corners were square, always 90*.  A job could be drawn, measured, and laid out more quickly and accurately. However, loosing those steps also changed the proportions. I have written about how this can see seen in vernacular housing design.**** I wanted to learn how an architect might have used the carpenter square. Robert Shaw was a good choice  because he wrote a book.

 

Robert Shaw's The Modern Architect was published in Boston in 1854.** 

 

 The pattern book's frontispiece shows the tools of the builder and the architect. The original drawing is an engraving which is quite dark. The color was added when the book was republished in 1995.

 

In the foreground is a large compass, probably used for stepping off. The architect holds a little one. The architect and builders are shown conferring, syncing the construction dimensions with the drawings .

 

 

Here is Plate 4, a 'Grecian  Frontispiece'

Where did Shaw begin his design?  Conceptually the design surrounds the door, giving it emphasis. So I began there.

Shaw himself stated that the door's height should be "...over twice the breadth of its height as three and seven feet."*** 

I have added the scale below the door: 3 units for the door's width. Then a half unit for the columns on each side and a full  unit for the width of the sidelights. 

These proportions follow those recommended by James Gibbs in 1732. ****

 

 

 

Was Shaw using 'circle geometry' for his layout? I don't think so. The circles don't offer much information. 

While the layout is 2 circles tall, the 12 points around the circumference of the circles give only the height, the width of the entry including the side lights, maybe the location of the transom. Note the arrows.

 

 

I think Shaw used a  simple geometric pattern that is derived from the circle, but which doesn't need to start with a length - a radius - and compass. It starts with the square which is easily laid out by the carpenter square. 

The width of the door and its sidelights was the dimension for a square. That shape was easy to lay out and make true with a carpenter square. Beginning with a length, he set up the corners with the square, added the lines for the 4 sides,  trued the box with diagonals. The diagonals used to find the additional height comes directly from the square. Done.        Note the arrows.  

Was there a name for it? Not one I've found.  It's basically a 'square and diagonal geometry'.

 

 The door, its transom, sidelights, and columns are also a square.

Here the quarter circle arcs, based on the width, cross at the top of the door frame, just below the transom. This layout, creating a slightly smaller rectangle within the square, was often used in layout and design. ****  I think here it is incidental.

I've extended the scale across the bottom and up the right side. It confirms the geometry.

The whole frontispiece is 8-1/2 units wide and 10-3/4 units tall. The door, the pilasters and the sidelights are 6 units wide; the columns are 1-1/4 units each. The columns' capitals are a half unit tall. The entablature is 2 units; the pediment, 3/4 of a unit tall.

Each unit and its parts could be stepped off with a compass. In 1854 the length could also have been stepped off in 12 inches intervals as marked on the carpenter square. As shown in Shaw's frontispiece in his book, it seems the builders used both.

 

The geometry used for the door and its parts is also used for the overall size: the height of the frontispiece is equal to the diagonal of the square.

The lightly drawn dashed line is the arc of the width of the door, showing how it lays out the square. This geometric proportion is also used for the sidelight glass panes (see the image above), but not those in the transom.

 

 

When we architects, restoration trades people, and historians note from visual observation that a particular building is Greek Revival, not Late Georgian, we are seeing geometry. I think we are recognizing, even if subconsciously,  that the rhythms, the proportions of  Federal architecture are different from the Greek Revival proportions shown here.   

 

* EAIA, Early American Trades Association https://www.eaia.us/

https://www.eaia.us/2025-rochester-ny

** Robert Shaw, The Modern Architect, Boston, 1854, originally published by Dayton and Wentworth, republished  (unabridged) by Dover Publications in 1996.

*** Shaw, The Modern Architect, page 63. 

****    For more information about James Gibbs' use of the door width as a unit of measure see: https://www.jgrarchitect.com/2025/01/james-gibbs-and-rockingham-meeting-house.html

            For more information about the square and its rectangle see:                                          https://www.jgrarchitect.com/2023/11/the-practical-geometry-of-parson_20.html

           For information about buildings using the 3/4/5 triangle for layout:

  https://www.jgrarchitect.com/2014/03/railroad-warehouse-frame-c-1850.html  

https://www.jgrarchitect.com/2014/10/the-cobblers-house-c-1840.html  

https://www.jgrarchitect.com/2013/10/1820s-farmhouse-north-of-boston.html 

  

Durer's alphabet via the book 'Good Eye'

A rift, not a book review

This is the cover for Good Eye, George R. Walker and Jim Tolpin's latest book about furniture design and proportion.*

The G comes from Albrecht Durer's alphabet in his  Instructions for Measuring with Compass and Ruler of his  Four Books on Measurement (Underweysung der Messung mit dem Zirckel und Richtschelettersyt, Book 3, published in Nuremberg, Germany, in 1525. 

Good Eye uses for Durer's letters for chapter headings. It discusses the design of the letters on pages 2-4. 

I last read Durer's book during Covid, 5 years ago. I wanted to understand Durer's knowledge and use of geometry.

Now I want to know more about his alphabet.

Durer used the numbers and letters of his time and place, the script of 1525 medieval Germany.The drawings I can find via the modern, online copies of his book are fuzzy.*  So his instructions are not clear to me.

 

I want to understand the basic geometric formation of the letters: Can I find out where did he begin? 

Here are Durer's letters A, B, C, D, F and Z.

 

 

 

I chose C and D.

I am laying out only the left hand C and D.

 

 

For both I began as Durer did, with a square and its center.  I have drawn these on graph paper to make the layouts easier to follow.

 

 

 

Using the arc of the square's side, here are the steps for dividing the square into 8 parts horizontally and vertically:

1) The square is divided into 4 smaller squares using a compass and straightedge.

2) the smaller squares is divided. 3) and again. 4) and again. That smallest width is 1/8 of the width of the square. The square can be divided into 8 equal sections both horizontally and vertically.

 

 

 

 

 

Here is the C. The circle within the square :

 

The second circle, its center moved one unit (1/8 of the width) to the right.

The circle cut at 1/8 of the width on the right side to create the letter C.

 

 

 

The letter D:        half the circle 

 

 

 

 

 

The second half circle with its center moved one unit to the left:

The leg of the D drawn one unit wide, set 2 units from the left side.

The serifs added: 1/4 of the circle whose radius is 1 unit wide. 

Look again at Durer's letter D to see the circles.

 

 

 

For more on Albecht Durer see: https://en.wikipedia.org/wiki/Albrecht_D%C3%BCrer  

*I have asked to borrow, via Inter Library Loan,  the 1977 translation into English of this book. It may be clearer. If so I will update this post after I've studied that.

    +                         +                          +                        +                      +                   +                        +

Good Eye is a good book for me, a geometer. It speaks from a different vantage point than mine. This is excellent. 

Here's why. The use of geometry in construction, including wood working, was passed down by master to apprentice, by hand.  I know of no teaching manuals for apprentices. From Vitruvius, (c.50 BCE,) to the 19th century pattern books, the writers assume the reader already knows how to use a compass, a scribe, and a line. We don't know what words they used to describe what they were teaching. We have almost lost the vocabulary as well as the skills.

Good Eye uses a different vocabulary to describe geometry. For example: the book uses focus for the center of a circle,  the point on which an arc pivots. I use Euclid's word: point. This is fine. I have watched George Walker explain geometry by sketching on a white board. I have seen Jim Tolpin's work and discussed it with him.  I know we are all exploring the use of geometry in design and construction. Good Eye helps me think more carefully. Thank you.

*Good Eye, George R. Walker and Jim Tolpin, Lost Art Press, Covington, Kentucky, 2024

James Gibbs and the Rockingham Meeting House

This blog post assumes you, the reader, are familiar with James Gibbs' architecture. If you need an introduction or a review, check the end of this blog. You will see links to what I wrote about him and his work. See also Wikipedia.

 

Did anyone in the States study James Gibbs' books?

Yes. Gibbs' On Architecture*, published in 1723, was imported to the Colonies. We know the steeple designs were studied and copied**. 

His book, RULES for DRAWING the several PARTS of ARCHITECTURE*, was also in the Colonies. 

Both books were in bookstores and private libraries. 

 

Were the rules Gibbs drew standard knowledge? Or was he simply the first to write them down? 

Did builders follow his layout instructions?  

I don't know yet. I'm studying historic doors, leaving surrounds and architraves for later research.

 

HABS has measured drawings of the Rockingham Meetinghouse in Rockingham, Vermont. It was  built from 1787 to 1797.  The Master Builder was John Fuller. The Master Joiner - who would have built the doors - is not recorded. He could have been John Fuller.

I know the Meetinghouse well. I've studied it, given tours, taught and written about the geometry of its construction as well as how the door paneling fits by the Rule of Thirds.**

 

The main door

 

 

 

 

 

 

 

 

 The HABS drawing of this door

 

 

 

 

 

 

 

 

That drawing with the dimensions inked out in order to make James Gibbs' geometry easier to read.

2 squares.

The width divided into 6 parts, 3 noted. Then one part (1/6 the width of the door) determining the width of the surround.  

I have used the arcs and lines that Gibbs used for his door layouts. The radius of the arc is the width and height of the square. This is a builder's 'shorthand'.

This layout matches the door on the left in Gibbs' drawing shown above.

 

A line can be divided into 6 parts using the Rule of Thirds. See Part II of my post on James Gibbs and the Rockingham Meetinghouse. The link is at the bottom of this post.**

 

 The door for the right stair wing at the Rockingham Meetinghouse

 

 

 

 

 

 

 

 

 

 

The HABS drawing for the right stair wing door

 

 

 

 

 

 

 

 

 The geometry: 

2 squares and 1/6 added to the height ( the red rectangle at the top)

This geometry matches the layout of the middle door in Gibbs' drawing of 3 doors shown above.

 

 

 

 

Then, I tried using the 1/6 part of the door width  as a radius.
I placed 3 circles on the width, the red line across the middle of the door. The dimension of the circles is the radius x 2: simple geometry.

 Beginning at the bottom of the door I stepped off 8 semi-circles up  the right hand side. They are the same width as those across the width of the door. Those semi-circles lay out the height of the door surround, the beginning of the architrave and its height.

Finally, I saw that the width of the pilasters on each side of the door was the same width as the circles. See the circle on the left pilaster.

The HABS drawings are small. The dimensions were made to record the building, not to record the geometry. Either the recorder or I may have missed nuance. This year, when the Meeting House is accessible, I will measure the doors to see how close what I've drawn is to the actual doors.

 

*James Gibbs,  On Architecture, 1728, London, Dover Press reprint

                         Rules for Drawing the several Parts of Architecture, 1753 edition through the University of Notre Dame  https://www3.nd.edu › Gibbs-Park-folio-18

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

       http://www.jgrarchitect.com/2022/02/james-gibbs-steeples.html

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

      https://www.jgrarchitect.com/2024/05/how-to-layout-pediment-350-years-of.html

A closer look at the Schiefferstadt House practical geometry

Please see my update:  https://www.jgrarchitect.com/2024/09/from-circle-to-pythagorean-triangle-via.html
I am not deleting this post because of my last paragraphs: I find the ways the Lines and layouts in Practical Geometry overlap need more consideration.* 

The lay out of the Schiefferstadt House,* uses a geometric pattern that was well known at the 1750's: the rule for drawing a square starting with a radius and a circle.

The diagram begins with the daisy wheel, scribed by a compass or a divider.  The 'petals' created by the 6 arcs of the radius around the circle make 6 points on the circumference.

The length of the radius for the circle is the width of the house.

 

 

 

When those points of the daisy wheel are joined they create Lines - dashed lines in the diagram. (Basic Euclidean geometry : 2 points are required to create a Line.)  The arcs of the radii cross those Lines to lay out a square. **

When one point of the divider, still opened to the width of the radius of the circle, is set on each of the 2 upper corners of the square, and the arcs swung, the arcs cross the circumference at the top of the circle.  Stepping off the radius around the circumference, will locate 6 more points.  All 12 points are equidistant from each other; all can be used for layout and design.

There is also a short cut to those upper 2 points; the place where the arcs cross the daisy wheel petals are points. 2 points = a Line. That Line extended is the same Line shown in my next diagram.  

 

The carpenter of the Schiefferstadt House could have used this geometry to step off  a rectangle about 18 units wide x 26 units long.  If his compass was open to a 2 ft span, the floor plan would have been 36 ft.wide  x 52 ft long. He would have trued his rectangle by checking that his diagonals matched, just as builders do today.

*

However the carpenter could also have used the square and its diagonal to lay out the plan. Those arcs would cross the circumference at the same place (dashed line), but they would cross the vertical lines of the rectangle about one unit higher than if the 12 points had been used (see the points where the dashed and red lines cross the circumference).  

This would make the floor plan 36 ft wide x 54 ft long. That's not much longer,  probably of little consequence to the design. However if the mason and the the framer were not using the same geometric progression (both using the first diagram or both using the second) the stone foundation and the interior wood frame would not have fit together. 

 

The drawings made c. 1978 for the restoration of the Schiefferstadt House may give me more information. The Frederick County Landmarks Foundation is sending prints. 

I will be looking for the simplest and quickest layout. I find that a builder tends to use the same same geometric progression for his plans and elevations. The geometry is one of his tools. The repetition of one pattern and one unit of measurement would be efficient and leave fewer chances for mistakes.  

If another layout is introduced it is usually the work of a craftsman whose work comes later - the finish carpenter adding a mantle, or the mason building a firebox and flu. Each might prefer a different system.

* The Schiefferstadt House, Frederick, Maryland, built in 1755, owned by The Frederick County Landmarks Foundation.  See my previous post for the geometry of the floor plan: https://www.jgrarchitect.com/2024/07/the-geometry-of-schiefferstadt-house.html

 

**2  basic practical geometry diagrams:

The diagram laying out how the radius of a circle can become the side of a square.

 Audel's Carpenters and Builders Guides , published in 1923, shows this diagram.

100 years ago, this geometry was common and practical knowledge.

The geometry of the Schiefferstadt House, Frederick, MD, 1758

The Schiefferstadt House 

 

 

This stone house, in Frederick, Maryland, was built c. 1755 for Elias Bruner, the son of German immigrants.

In the spring of 2023 the board of the Preservation Trades Network toured the house when we were in Frederick to plan the 2023 International Preservation Trades Workshops. The guides of the Schiefferstadt House showed us the house from cellar to attic, sharing both the original construction and the on-going work of maintenance and restoration. 

Our visit was well worth our time; many thanks to the Schiefferstadt staff and volunteers.

We saw many practical built-in systems for cooling and heating. 
The black blob in the lower left corner of this picture is one of several vents for the very effective basement cold storage vault. 

Under one kitchen window is a very useful drainage sill. 

 

 

 

 

 A close up of the trough with its lip and spout.  

The kitchen help need not carry the dirty dish water over to the door; instead it can be efficiently discarded out the window. 

 

I have seen a trough like this only once before, in medieval military barracks
in Switzerland.

The HABS drawing of the first floor plan shows the original stone house at the top of the drawing. The brick wing was built later. 

That first, 1637, house is outlined in red.

Stone walls are usually built between 2 lines, set to keep the walls straight. Here either the exterior line or the interior line could have set the governing dimensions.   

In both cases the practical geometry of the Line and its arc created the plan. First: The exterior geometry:

Using the width of the main house (red arrows) as a radius for a circle, I drew the daisy wheel with its 6 'petals',  noting the points around the circle where the arcs cross the circumference. The points connect the lines which lay out the long walls - see the red dots and vertical dashed lines at the top of the drawing.

The arcs of this daisy wheel create 6 points  If I rotated the wheel to begin the arcs not on the corners of the house, but at the center of the lower wall, the petals will be perpendicular to the side walls, parallel to the front and back walls. This adds 6 more points to the circumference - noted here as black dots, 12 points in all equally spaced*.

 2 of the new points lay out the location of the 4th wall, here at the top of the drawing, noted with red dots and a horizontal dashed line.

 

However, the plan for the house could also have been laid out from the inside. The exterior sides of the stone foundation walls could have been irregular, sloping away from the house below grade, only becoming straight  once they were above grade. I have seen this often in houses built before 1900. As in the first layout the width is the governing dimension. The geometry, a square and its diagonal, lays out the interior walls.

The builders dug into the side of the hill to set the foundation. When they set their governing lines which layout did they use? The exterior or the interior plan? Perhaps the exterior plan was concept, drawn during a consultation with the owner, then staked on site, perhaps with offset lines like those we use today. Then the interior dimensions could have been used when the workmen were on site, in the future cellar.

On the drawing are faint pencil lines of diagonals and an arc, using the exterior width as the side of the square. They don't quite fit. They are an exploration,  an essential part of discovering what geometry the builder used.

 

 

The house has a center entrance, a room on each side. Within both walls on either side of the center hall are fireplaces and flues. Here is the kitchen fireplace.

 

 

 

 

 Fireplaces on both sides of the center hall itself send heat to iron boxes on the first and second floor. This picture is a 2nd floor heating box, set into the wall between 2 rooms so it will radiate heat into both rooms. It's an early version of the radiator.

 

The flues of those first floor fireplaces join to become one chimney as they exit the roof.

Here 2 members of the Preservation Trades Network board and a staffer for the House stand in the first floor hall under the arch created by those flues as they come together overhead. One fire box is visible on the left. The one on the right has been closed.

This 3D drawing of the House is helpful even if not quite accurate. The flues come together over the first floor, not the second  as shown here.

However it explains how 'wishbone' masonry chimney blocks were located and the flues joined to become one chimney at the ridge of the roof.  

Each wing of the chimney required its own foundation which would have been built as the stone walls of the house were laid up.

I found as I studied the layout that while I could easily layout the exterior dimensions, I did not have enough experience with stone construction to understand how the masons would have worked once they set up the lines for the foundation walls. How did the masons measure where to set the chimney foundations?

A timber framer can begin the house frame with a sill set on the foundation after it is complete. I know those frames and foundations well. The joist pockets for the wood interior frame of the Schiefferstadt House had to be set in the stone as the wall was built. How did they know where to place them?

I asked Joe Lubozynski for help. He lives in the area, is an excellent architectural historian as well as craftsman. He knows historic stone construction and this house in particular. And he was willing to advise me. 

He and I reviewed how the house would have been built, beginning with digging into a slope (see the Architectural Cross Section above),  placing the footings, laying the stones for the walls and chimney foundations.  Then constructing the basement cold storage space with its vaulted cellar ceiling and placing the first floor frame.  

We considered whether the builders would have measured from the outside or the inside of the stone walls to place the joists. Our conclusion was obvious: it's much easier to set Lines from the inside of the walls and check them. The work would be done more accurately as well as more quickly.

At this point we drew the Rule of Thirds* diagram using the inside of the stone foundation as the rectangle.

 

  
The diagram easily located the floor joists. Note  how the inner sides of the floor joists are on the intersections of the diagonals. the diagonals cross, making 2 points. The points layout out the location of the joists and the joist pockets. See the 2 black lines with arrows on each end.The geometry is as simple as the layout of the foundation.

The chimney bases are white in this drawing in order to make the lines of the Rule of Thirds clear. 

 

 

The Rule of Thirds also locates the 2 foundations which support the fireplaces and chimneys. See the red arrows.

The measured drawings were done before the restoration of the house began. The fireplaces had been altered. The location of the masonry for the fireboxes and chimneys is an educated estimate. I have not measured the rebuilt and restored masonry. 

 

My conclusion? The technology of the Schiefferstat House is elegant and sophisticated for that period in the Colonies. The grand houses in New England and Virginia did not have the conveniences of this little house.  However, the practical geometry of the house is simple, traditional: very similar to what I've found from the same era in New England, New York, Virginia, and Louisiana.

* For information about Daisy Wheels,  drawings squares and rectangles with compasses, the Rule of Thirds, try:

https://www.jgrarchitect.com/2020/04/practical-geometry-lesson-4.html

https://www.jgrarchitect.com/2022/11/virginia-folk-housing-update.html

 https://www.jgrarchitect.com/2022/10/serlios-lines.html