Jane Griswold Radocchia


Architect / Geometer / Historian




Jane Griswold Radocchia



Architect - Geometer - Historian



Archive - 2022



Thursday, April 21, 2022


The Parson Capen House, 1683, Topsfield, MA

This post, first published in 2014, has been revised based on a better understanding of the geometry.

Parson Joseph Capen built this house in 1683 in Topsfield, Massachusetts. He ministered to the town from 1682 until his death in 1726.

The story about this house: It is still here because it was owned by an old Boston family with extensive land holdings.Their herdsmen drove cattle to market in Boston along Rte 1 (which is practically next door to this house) and used the house as a way station on their trips into the city. So it wasn't torn down, updated, or abandoned. We are lucky.

The pictures are from the LOC HABS archive. Note the drops and brackets. The windows are casements; in the picture they have been swung open.

For more information and photographs in color see http://www.topsfieldhistory.org/parson_capen.shtml

The geometry for the house is based on the square.

Here is how a square is derived using a compass, a straightedge and a scribe.



Once a carpenter knew his geometry he could layout a square in fewer steps.

The square with its arcs gives the carpenter 4 points - where the arcs cross each other - for dividing his square in half horizontally or vertically. This could have been used for the hall and the second floor beams.



The chimney and fireplaces were to be in the middle of the house.

So the builder laid out the house foundation from the chimney block.

The diagram shows in the center the brick which would have formed the back wall of the 4 fireplaces with 2 square spaces on either side: the parlor and the hall.



Next comes the fireplaces themselves, on both floors, and the flues.

On the first floor the main masonry block is a square in plan, the oven needed half a square. Its flue joins the hall chimney.



On the second floor the chimney mass is square.



The house has 4 bents, one on each end and one on either side of the chimney mass. The fireplaces depths on the second floor determine where the 2 interior bents are placed.

It's possible there are bents across the rooms as well. However, since the summer beams on the first floor appear to be located over windows and the second floor beams do not match the first floor beams, I think not. The Rule of Thirds was used to lay out the bents. It determined the heights of the floors and the placement of the windows. (The red arrows indicate ceiling heights.)



See this description for how the Rule of Thirds works: https://www.jgrarchitect.com/2020/08/lesson-6-rule-of-thirds-part-1_21.html

The cantilevered end beams for the second floor have drops below them. The beams for the interior bents have corbels. These are quite visible in the HABS photographs. The cantilevers for the roof also have drops, as well as corbels in the middle of the roof overhangs on the end elevations.



The second floor overhangs the first floor by about a foot on the south/front side. This was popular in England as a way to protect the daub and wattle walls from rain and wind. Here weather boards - known today as clapboards - covered the frame, but the tradition continued. The attic extends out over the second floor on the sides of the house for the same reason.

These HABS photographs are beautiful and clear. Click the images to enlarge them.



The roof pitch and the placement of the ridge pole might have been laid out from the second floor. I have seen this proportion - the crossed arcs of the side of the square - in other First Period houses, It may have been used here. I am currently exploring how a 12 pointed daisy wheel may be the base for the layout. I'll add more as I learn.

Note the corbel beside the 2nd floor window which supports the roof overhang.



Here is the front elevation with the drops and corbels noted. They accent the ends of the cantilevered beams which are the top plates of the bents.



The beams:
In the hall the beam which supports the second floor joists was set in the center.

The parlor, the room to the left, is larger. It needed 2 beams. So the space is divided into thirds. These beams are joined to the beam that runs between the 2 bents on this side. The windows were placed where a post would be located under those beams if they were part of bents.

On the second floor the ceiling beam are centered. All the beams appear to be set to the side of the lines, not on the line.



This drawing may be an accurate depiction of the front elevation. However, the plans are not quite consistent with this layout. The windows might be set equidistant from the corners of the house, or not. They may be centered on the second floor rooms, but not on those on the first floor.

Both sets are grouped together in the same geometry. The casement windows are all the same size.



The measured drawings for the Historic American Building Survey, HABS, were done at 1/8"= 1'-0", a scale which is fine for concept, but not good enough for serious consideration of construction details. They have very few dimensions. The drawings from 1916 do not quite agree with HABS. Some observations:

  • *The Golden Section is not used here. I find that the Golden Section is about growth; houses are about stability.

  • *The front door is not centered on the facade; if it were the door could not be opened back against the front wall. The brackets sit under the 2nd floor beams extended to support the cantilever.

4/21/22: I wrote much of this 8 years ago. The layout of the foundation based on the location and size of the chimney back still makes sense. I revisited the framing and the elevations, understanding that the layout begins with the framer who must decide where the bents will be; how tall; where the marks for the mortises and tenons will be. And how will the second floor and roof cantilevers be supported?



Friday, February 25,2022


James Gibbs' steeples

James Gibbs was the Surveyor of the Work for the design and construction of St. Martin in the Field, Trafalgar Square, London, begun in 1722, completed in 1726.



His pattern book, On Architecture, published in 1728, had 150 plates. 7 were engravings for St. Martin's. He writes that Plate II is "The Geometrical Plan of the Church and Portico, shewing the Disposition of the whole Fabrick." (Introduction - i) Plate III, shown here, is "The West Front and Steeple"


Many churches and steeples are included in Gibbs' book. Plates 29 and 30 show 6 images of steeples, all drawn for St. Martin's but not chosen. Plate 31 has 5 draughts of steeples for St. Mary le Strand.

In 1775, the Providence (RI) Gazette, writing about the Baptist Meetinghouse, comments on the use of the "middle Figure in the 30th Plate of Gibbs designs" * for the church steeple.


This engraving is the draught (the architectural drawing) of the Geometric Plan of that steeple. Gibbs writes that while steeples are Gothick, "...they have their Beauties, when their parts are well dispos'd, and when the plans of the several Degrees and Orders of which they are compos'd gradually diminish and pass from one form to another without confusion, and when every Part has the appearance of a proper Bearing." (viii)

The Master Builder for the Baptist Meetinghouse was Joseph Brown. How did he know what to do from those instructions?

He was not only a builder but an astronomer, a scientist and a professor. He knew his Geometry.

How would the parts be 'well dispos'd' or well ordered. That could refers to the pattern of 'base, column and wall, architrave' for each section.

Or it might be how the parts are all the same height. I have marked on the engraving where each part begins and ends. Each provides physically and visually 'a proper Bearing' for the next level.


How did the parts 'gradually diminish'?

Below each steeple on Plates 29, 30 and 31 is a cartouche, a diagram: the plan for each steeple, showing the outlines of each steeple part.

This is the diagram for the middle steeple which was copied for the Baptist Meetinghouse.The image in the book is 1.5" square. It is the size of the image Joseph Brown, Master Builder, would have worked from.




I have labeled the outlines of each part of the tower to correspond with my numbers on the steeple drawing above. (5) is the base of the 8 sided spire. The innermost circle is the cap where the weather vane is attached.

The parts layer one on the other following a diagram, a pattern - which I refer to as the square and its circle. This geometry was well known. It goes back in construction to at least the 7th c. in Constantinople. Serlio placed it on his frontispiece.**



This variation of the square and its circle, uses only the diagonals and adds the division of the square into quarters. As the design has 8 sided Parts (#3 and 4) and an octagonal spire, perhaps this diagram was used.



Where the Lines cross the square locates a smaller square, rotated. And those Lines locate the next. They determine the size and location of each Part of the steeple

The diagram is not meant to be a working drawing. Instead it directs the builder. It does not matter if it is not quite accurate. When the builder lays out the work he will adjust and refine the shapes to fit his frame.



What happens when the first square is rotated - creating an 8 pointed star?


And the squares that fit inside that square are added? Drawn here in black over the first diagrams drawn in red.

The octagons of the Parts are laid out. Drawn on a framing floor the lengths for each wall would be easy to measure and set correctly.

It's done with just a length of twine and the knowledge of geometry.


The Providence, RI, Baptist Meetinghouse - with its steeple, drawn c. 1800. The image is now in the Library of Congress with the HABS drawings of the Meetinghouse.


* quote from American Architects and Their Books to 1848, ed: Hafertepe and O'Gorman, UMASS Press, 2001, Abbott Lowell Cummings' essay, The Availability of Architectural Books in Eighteenth-Century New England, p. 2.

** the Geometry of Hagia Sophia's dome (Bannister Fletcher's diagram)



The lower right corner of Sebastiano Serlio's frontispiece of his On Architecture: a cube with its diagonals, the circle and the next square that fits within that circle. As can be seen in the steeples drawn by James Gibbs, these circles and squares can grow in and/or out.



Tuesday, January 11, 2022


James Gibbs' Of Architecture,
Draughts for a Menagery,
Part 2 of 2

Part 1 of 2, the post for Gibb's design of the Menagery which was built is here:

James Gibbs' On Architecture included 2 options for a menagery. In Part 1 of 2,  I wrote about the one which was built at Hackwood Park, an estate near London. It is still standing.
Here is Gibbs' portrait with his compass, the mark of his profession.
Gibbs' expected "any Workman who understands Lines" * to be able to execute his designs. What would a workman have read in Gibbs' drawings? What could I read? Do I understand Lines?



This 'menagery' was to be a welcome destination for those strolling through the Hackwood Park estate grounds. Built of stone, the menagery would not have been as dark as this image.The 'draught' elevation accentuates the quoins and articulated arches, to give the workmen the necessary information.


Like the design which was built, the pavilion required a gracious porch with a room on each side: one for serving and drinking tea, one for the quiet perusal of books about nature, especially birds. The living quarters for the staff who took care of the estate's pheasants were around the back.


The floor plan begins with the central form: the porch and caretaker's quarters.  Note that the red lines for the left and right sides of the rectangle are located at the back of the fireplaces. Perhaps this is information for the mason: he will start his work there. 

The depth of the building (the red line of the right side) is the radius for the arcs which cross at the wall on the left side. I have drawn the left line that completes the square that the arcs determine. The intersection locates the left wall noted by a red line with arrows.


The shape comes from the circle and its daisy wheel.  It is a rectangle created by the 6 points of the daisy wheel on the circumference.

In case the diagram is not clear I have added notes.

Begin with a circle and its daisy wheel.

The radius of the circle always steps off around the circumference 6 times: 6 points on the circumference, 6 daisy petals made by the arcs of the circle.




Connect 2 points (solid black line). This will be the length (front to back) of the central form. Add the Lines ( dashed black lines ) perpendicular to the first Line.



If the central form were to be a square, where the dashed lines cross the daisy petals mark 2 more points. The dashed, black vertical line is the 4th side of the square: each side is the length of the radius.



This central form is not that wide.  James Gibbs choose to use the 2 vertical points of the daisy wheel to locate the Line for the left side of the porch and the caretaker's quarters. The black box is that rectangle.


The rest of the layout is based on the interior space. Gibbs assumes the workmen who might copy his 'draught' know how to build walls; he is not providing a construction document.

The wall between the porch and the living quarters is set beside the center of the rectangle. This makes the porch a little more gracious. It is also the way a mason sets lines today, building beside his lines.


To determine the location of the wings to the main body of the menagery Gibbs divides the rectangle into 8 parts.

Here is the diagram showing how to use Lines to divide a rectangle into 8 equal sections.

I call this 'The rule of Thirds' because artists use this diagram as a design tool and that;s what they call it.  It's the 3x3 pattern that appears when we edit cellphone pictures. That pattern is 1/3. Here the division is into 1/2, 1/4, and 1/8. **


The wings of the menagery are set back 1/8  the depth of the main block in the front and the back.  I've drawn the open porch divided into 4 equal parts and extended the Lines to show the depth of the wings' setback. This offset is used on both wings, front and back.


The wings are themselves both 3/4/5 rectangles. They are square: a 3/4/5 triangle always has a 90* corner. The diagram shows how to layout a 'right triangle' using Lines.



The 3/4/5 rectangIe was a common way to add a wing to an existing building. If the mason set his length against the central form at 4 units and his width at 3 units, his wing would be square against the main block. His stone work would be true.  I have drawn the 3 x 4 units here. I have also left my pencil marks for further information (enlarge the drawing!)


The layout for the 3 arched openings on the pavilion's porch also uses the Rule of Thirds.

Here is the diagram above turned 90*. Note where the red lines cross the
black lines above my red dashed line.  Those points divide the square into thirds. Drop a vertical there (see red lines above)  to mark the columns' center line.


While this menagery design is more complex to write about than Gibbs' other design*, it is quite easy to lay out with a compass and straight edge. A trained workman would have known the steps, some of which, for example the 3/4/5 rectangle, are used today.


 The elevation?It's4 squares and the same pediment layout that  Gibbs used on the menagery design which was built.


 Gibbs began the elevation from the same inside dimensions he used for the floor plan.
A note: the windows are centered on the rooms' inside wall, but not on the exterior width of the wing.  The quions are such a strong visual vertical that they appear as an anchor. The windows were centered on the rest of the wall.


The pediment is drawn  following the rules described by Serlio. For step by step instructions, refer to Part 1 of 2 of this post of the draughts for the Menagery.*


* Gibbs' book On Architecture, published in 1728, includes 150 plates: plans, elevations, sections and perspectives of buildings Gibbs had designed and built. The quote is from his introduction, page i. A reprint is available from Dover Publications.


** I've posted about The Rule of Thirds in more detail here:




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Thursday, April 21, 2022


The Parson Capen House, 1683, Topsfield, MA

Friday, February 25,2022


James Gibbs' steeples

Tuesday, January 11, 2022

James Gibbs' Of Architecture, Draughts for a Menagery, Part 2 of 2