Monday, June 29, 2020

Practical Geometry Lesson 5, Addendum

Why I left out diagram K from Owen Biddle's Plate 1 in his Young Carpenter's Assistant.

Lesson 5 was written for a student today who wants to draw rectangles using practical geometry.
Biddle was writing for the apprentices he worked with in 1805. They needed to know the practical application of geometry for the buildings they worked on - including the curved parts.

This addendum is like one of those long footnotes in an historic report -  a part of the story that's not quite germane to the subject, but ought to be included.

Biddle  identifies each diagram on Plate 1with a letter. There is no diagram for D. However, in his text, between C and E he discusses the mathematical instruments a carpenter should obtain. Perhaps this is D.  I quote him:

- scales of equal parts on the thin ivory or box rule
- a bow pen or compass
- a small piece of gum elastic for rubbing out black lead lines
- a stick of Indian ink
- 2 camel's hair pencils, one large, one small
- a black lead pencil

There is also no J. And there is no text in its place as exists for D. 

Here is K.  

Biddle writes: "Three points (not in a right line) or a small part of a circle being given to find a center which will describe a circle to pass through the points or complete the circle."

                                                     Start with a curve a-b .
                                 The curve in Biddle's drawing above is a-b-c.                       

 The curve divided in half:  Swing 2 arcs that are the same length  above above and below the curve: a-c and b-d. Mark where they cross, at f above and below the curve,

Connect  f and f with a line - here dashed. Mark where the line crosses the arc a-b -  I've labeled it g.
This line divides the arc in half. 
If 2 lines were given - here: a-g and g-b , this step would not be necessary. Biddle's diagram  labels his lines a-b and b-c.

Now, the instructions become complex.
Draw it step at a time. And consider that this is only Plate 1 of Biddle's pattern book. He included 43 more Plates for the carpenter's assistant.  

Divide the lines a-g and g-b in half.
This is shown in Biddle's E  and F diagrams. Check Lesson 5. 

Extend the lines which divide  a-g and g-b in half so they intersect at k,
K is the center of the circle which passes the points or completes the circle.

Refer to Biddle's drawing K above for the complete solution, all neatly explained in only one diagram.  

Clearly Biddle thought this information  was essential knowledge for  every carpenter. His next Plates illustrate why. The construction his 'young carpenter's assistant' would be working on involved determining and laying out many curved lines for vaults, arches, windows, stairs and railings.

Plate 2 discusses ellipses: how to draw them using geometry or a trammel, how to find the center and axes of one already drawn.   

Plate 3 is concerned with octagons, arches, groins. the use of trammels, how to divide a line into parts.

I am quite fond of Figure 1, describing " an Octagon within a square." . Simple, quick, even obvious - if you know geometry.

I have seen  painstaking explanations of  how to lay out an octagonal using algebra: quite painful.

Plate 6 reviews raking cornices and "the sweep of a cornice which will bend around a circular wall and stand on a spring."

Plate 31 lays out "the section and elevation of a circular or geometrical stairs". Biddle includes in figure C  "the manner of drawing a bracket for the ends of the circular steps..." and the careful, detailed instructions.

Plates 32-35 - not included here - explain how to layout the newel, the falling moldings, the hand rail for such a stair.

Biddle's Young Carpenter's Assistant, Owen, Biddle, 1805, originally published by Benjamin Johnson, Philadelphia, and Roland and Loudon, New York. Reprint by Dover Publications, Inc. 2006. If you want this book, you can easily order it from them directly. It has an excellent 15 page introduction with bibliography by Bryan Clark Green.

Tuesday, June 16, 2020

Practical Geometry Lessons, Lesson 5: Rectangles

Today these skills are not required knowledge for builders. We have steel carpenter squares that have true 90* corners, as well as levels and  lasers. 

The carpenter squares shown here are some of the earliest made in the States. They were made in Shaftsbury and N. Bennington, VT, 1825-60. Some are on display at the Bennington Museum; all are available for study.

The 1503 woodcut at the end of this post includes a square being used for a layout. That square might not have matched the square of another builder.

Practical geometry taught how to 'prove' that an angle was 'true'. Carpenters today still make their work 'true'.  

This "Geometric Problem'  and its solution was particularly important when carpenter squares were not necessarily true: the square corner was not always accurate, not dependably 90*.

This is the end of Lesson 5.
The carpenter's assistant who masters these problems is now ready to assist in layout and framing. Maybe he (no recorded 'shes' that I know of) will go on to learn design. 

For more ways to draw a square  see Drawing a Square, Parts 1 and 2.
Part 1:
Part 2:

After-thoughts and questions: How did carpenters carry a 10' rod?  Did the rod fold, or come in pieces that could be connected with pegs,  perhaps leather sleeves?
I have seen one in a medieval print, part of which is shown here.

The builder on the left uses a square for layout. Below him is an axe, a saw, and a level with a triangular hole and a plumb bob.
The builder in the center holds a 10' rod.   The landscape to his right is the site where he will layout a building, or perhaps land.
 A 'rod' when used in land surveying is 16.5 feet long. It is also called a 'perch' or 'pole'. How did they carry that awkward length? 'Links' and 'chains' are also used in surveying, so perhaps a rod could be a length of chain.

 Woodcut by Gregor Riesch, Margarita Philosophical, published in Baden,1503.

Sunday, May 31, 2020

Owen Biddle's "The Young Carpenter's Assistant"

 Owen Biddle's pattern book The Young Carpenter's Assistant was published  in Philadelphia in 1805.
I am rereading it, looking especially at his use of Practical Geometry.

His first drawings, A and B, on Plate 1 are a plan and section of a drafting board. His instructions include how to construct the frame and the panel, and how to locate the small wood buttons which will keep the dampened paper in place while it dries by the fireplace to become "smooth as a drum head" .*
Drawing C is a T Square whose construction is also described.
He lists the tools a student would need; including a small compass, often called a 'bow pen'.

The rest of Plate 1, Plates 2, 3, with Biddle's descriptions explain "the most useful geometric problems every Carpenter ought to be acquainted with."*

I read between the lines: What was common knowledge that did not need explanation?

- 'Carpenter's Assistants' might have been men who needed to learn more about construction but had not had regular apprenticeships. Many apprentices ran away before completing their training; the most famous one is Benjamin Franklin.
 -  The carpenter's assistant had little experience with paper for architectural drawings. He used a convenient board, a compass, a straight edge, and an awl when he needs to work out a design. These designs are often found on boards during renovation.
-  Biddle specified the compass to be small because there are many different sized compasses. The ones shown here are in a cabinet maker's shop. The biggest is about 16" tall
-  Geometry was a practical tool in construction.

Biddle had set up a school to teach 'architecture' to carpenters.  Assistants might not have been trained to draw in their apprenticeships. Those who designed might not trained under a master builder.  Biddle himself was a contractor becoming an architect.

Biddle has  includes 2 drawings of the same construction. Biddle first explains the design, Plate 15. Then he details how to make the construction drawing become an inviting illustration for a potential client, Plate 16.

Here is Plate 15. In his description Biddle says he gives 'the lines of a pitch pediment frontispiece'.  He writes about 'Lines' as Serlio did: the Lines for layout and design. He adds that the  "Column is made 10 diameters in height." The use of diameter of the column as a measurement is discussed by Palladio.

Biddle says for the best appearance "the door should be as wide as  half its height"*. The scale on the right side of the drawing is labeled '10 diameters'. Did the 'door' consisted of just the part that's hinged? No, Biddle's dimensions include the door and its surround. Here is the door with a height of 10 units divided into 2 red squares, each  5 units square.

As I followed Biddle's notes and dimensions on the drawing I learned how he thought about construction and design: his unit is the door and its frame: the 'door'  itself was not the measure; the door frame between the columns is. The column's height includes its base and its capital, but not the lower base  which is noted as 8" high. This allows the base to be adjusted to fit the vagaries of the location.

Of course the fanlight is a semicircle. The frame is half the height of the squares of the door. 
The key stone of the semicircular fanlight was the center of the square, the height of the pediment 3/4 of the square.

The roof pitch falls exactly along the Lines which cross the square and locate its 3/4 mark.

The notes on the right side about the height of the little columns in the architrave and the  return of the eaves reference the span of the door which is derived from the height of the columns.

Biddle notes at the bottom of the actual door its width: 9 parts.  the parts lay out the panels. The panels themselves are 3 parts wide. Their length and spacing is also determined by the 9 parts.
The fan light's rays are also set by the 9 units. . The tic marks for the 9 units extended to the arc of the fan light layout the fins. The rhythm for the main fins is 1,2,3,2,1.  In the center of the 'petals' of the fan lights are more tic marks - the centers of the arcs that create the tracery.

I tried to draw this clearly. What resulted was a jumble of lines. I'll try words:
Biddle began with the diameter of the column. He used 10 diameters to create 2 squares for door and 3/4 of the square for a fanlight and architrave. His 5 unit width is divided into 10 units, 9 of which are the physical door. The Lines inform all his design.
Biddle uses 10 parts because he supposes
"that the door is for a town house with a narrow front..."  The parts will be in proportion to each other, but take up less space.

The  square shown here has already been divided into 4 equal parts. The Lines divide the 4 rectangles into 5 rectangles. The bottom and top lines defining the square are thus divided into 5 equal parts.

Plate 16

Here is the same entrance shaded. He writes " ...the student should make it his business to understand the effects of light and shade..."  He explains how to think about shading in 2 paragraphs, where "the shade should be strong", where the part should be "bright".

Plates 17 and 18  are similar: one drawing setting up the design itself, the next explaining how to shade the design.  

* Plates 1, 2, 3, 15, and 16, and their descriptions
Owen Biddle, The Young Carpenter's Assistant; or, A System of Architecture Adapted to the Style of Building in the United States, 1805, Benjamin Johnson, Philadelphia, and Ronalds & London, New York. Reprint 2006, Dover Publications, Inc.

2 good biographies of  Biddle: 1) the introduction to the Dover Edition of Biddle book by Bryan Clark Green. 2) an article by Micheal J. Lewis in  American Architects and Their Books to 1848, edited by Hafertepe and O'Gorman, 2001, U.Mass Press.

Tuesday, May 12, 2020

The Geometry of The Park-McCullough House Brackets

I have not measured nor carefully checked the brackets for their construction. The House is not open during this time of the corona virus. I will go another time.
The architect's perspective drawings of the House are part of the House' collection. I am curious to see if the bracket design is visible, and whether working drawings for the house exist.
I have heard that a mock-up of the bracket may exist. By June I hope to examine it.

I would like to understand how the design for the bracket evolved.
The shape was well known. It is one variation of the standard shapes used in classic architecture. Here it is in Plate I from Asher Benjamin's The Country Builder's Assistant, 1797, Greenfield, MA.  F is a cima recta molding; D , E, I and K are the reverse: ogee moldings. (and E have quirks; E has an astragal. K has a bead.)

Did a joiner suggest the Park-McCullough variation to the builder, the architect? Or did the architect, possibly his draftsman, draw it for the others?  Did they build a mock-up and improve upon it?

 Here's Edward Shaw conferring with his builders, from the frontispiece of his pattern book published in 1854, 10 years before this house was built.

I wrote blog posts about him here:
and here:

I've written about the House and its Barn being 'built to the weather'. Start here:

Tuesday, April 28, 2020

Practical Geometry Lesson 4b - The Old First Church

The circle, the daisy wheel, governs the design and frame for the Old First Church.
The circle was often used for the top of the window in the 19th century as well as just for decoration.  
I am sorry the images are tinged with green. This is one more presentation skill I have not quite mastered. Another proof reading is also necessary.  

Friday, April 10, 2020


The drawing of 'stepping off' is from Audels Carpenters and Builders Guide, NYC, 1923, p.640.

The holes in the daisy wheel are lower left, center, upper right.  There are others which do not photograph as well. This sheathing board was given to me by the crew that deconstructed the barns. It travels with me when I present on Practical Geometry.