Saturday, October 29, 2016

Practical Geometry, Drawing the Diagrams #2, the 3/4/5 Triangle

Here's the second diagram I taught at the 2016 PTN Workshops.

I did not lay it out as I have done here. Today I think this diagram would have been a good handout.I could have drawn it; the participants could have followed along and had a cheat sheet to take home.

Using the  3/4/5 triangle for construction

 3/4 5 triangles always have a 90* angle where the side with 3 units meets the side with 4 units.

Draw a line and mark off your unit.

Lay out lines of 3 units, 4 units and 5 units.
On my diagram:  A-B = 3 units
                            A-C = 4 units
                            B-D = 5 units

Swing an arc from either end of A-B; one arc with a 4 unit radius, one arc with a 5 unit radius,
Where the arcs cross is E.

Draw lines from A to B  to E to A.
This is a 3/4/5 triangle. The corner at A is 90*

For fun I have laid out another triangle beginning with 5 units, use 3 and 4 units for the radii of the arcs. It is another 3/4/5 triangle with a 90* corner.

We used Gunston Hall, built of brick by George Mason from 1755 to 1759, as an example. Mason  was a real mason; he gave George Washington advice about mortar recipes. He would have used the 3/4/5 triangle when he built walls or, as a Master Mason, instructed others. The triangle was/is one way to keep brick square and true.
It would have been ordinary for him to use 3/4/5 geometry to design his house.

The base of the brick work at Gunston Hall is 4 units. The height of the brick work of the end wall at Gunston Hall measures 3 units. The diagonal is 5 units.

The floor plan is also laid put using the 3/4/5 triangle. See my post for more information and drawings:

I asked the participants at the PTN session to divide the width of the Gunston Hall side elevation into 4 equal parts. I wanted them to draw the geometry for themselves, to see it come to life.

Again a handout with step by step instructions would have been helpful.   
Not everyone knew how to divide a line into parts; but those who did showed those who didn't. It was a excellent group.

One of the first figures in the pattern books on Practical Geometry is the division of a line by a perpendicular. Here is  Figure 5, Plate II, of  Asher Benjamin's The American Builder's Companion, first edition published 1806.

 Asher Benjamin's Figure 3, Plate II,  shows a
simply drawn 3/4/5 triangle expressed with units 6/8/10
with short arc lines at c, the top, to show the use of a compass to make a circle with the radius determined.

His description assumes a familiarity with the language of geometry and compasses.

"To make a perpendicular with a 10 foot rod. Let b a be 6 feet; take eight feet in your compasses; from b make the arch c, with the distance ten feet from a; make the intersection at c, and draw the perpendicular, c b. "

Thursday, October 6, 2016

Practical Geometry - drawing the diagrams

The participants at the hands-on sessions I taught on Practical Geometry  at the 2016 PTN Workshops, asked me to post the diagrams for the basic geometries they worked with.
Here is the first.

How to divide a square into thirds:

We used graph paper for the first geometry so everyone could see the lines develope into a pattern. Everyone could count the squares to be sure they were following directions.

1.  Draw a square 12 units wide and 12 units long. label the corners A, B, C, D.

Add the diagonals - the lines from one corner cross the center to the far corner. A to C; B to D.
The lines will cross in the center of the square. Count the units to prove this to yourself.
Label the center of the square E.

Divide the square in half vertically, F to H - follow the line the graph paper.
Divide the square in half horizontally, G to I - follow the line on the graph paper.

This is the basic pattern. The square can now be divided into 3, 4, or 5 (or more)  equal rectangles as needed.

2. To divide the square into thirds:
Add a line from each corner to the middle of the opposite side. A to G and B to I.
These lines cross the original pattern.

K and L, if extended parallel to A-B, would define a rectangle that is 1/3 of the whole square.

A rule in geometry is that there must be 2 points to establish a line.
 Below is a diagram of  how the diagonals from the corner of a square to the middle of the opposite side give 2 points for the lines which divide a square into three rectangles of equal size.

This division of the square into thirds is often found in pre-Industrial Revolution design.
I do not think framers drew out the whole diagram on a sheathing board or a framing floor. Rather because the diagram was common knowledge they just drew the parts they needed.

An example:
At the workshop I taught the application of this geometric pattern using the plans and elevations from a cabin at Tuckahoe  -

The cabin  is very similar in size and dimensions to the slave cabin at Clermont Farm which was across the way from where we were meeting. 

The end wall of the cabin is  2/3 of a square. The roof  begins on the 2/3 line. Its pitch follows the diagonals of the upper square. the windows, doors and fireplace are centered on the square. That's all.

I then showed the group how Owen Biddle used the same geometry to tell a mason where windows and doors were to be placed.
The  elevation and floor plan are both composed of 2 squares. On both the window placement is one side of the center line. The  diagonals from corner to center call out the window width ( on the elevation and the interior partition on the floor plan.
In the floor plan I have used a dashed line to note the lines dictating the window width.

I did not show them how master joiners layered the squares divided into thirds over each other to call out the dimensions and relationships  between parts of doors and architraves for Georgian Meeting Houses.
Shown here is perhaps how the the main door for Rockingham Meeting House, Rockingham, VT, may have been laid out.
 As I look at it today I think the diagram may be too complex - time to look again.

There is no record of what the carpenters and masons called these geometries. They would have have been explained verbally while taught by doing, never needing to be recorded.
There also may be a notation system that we do not recognize  - yet. .

Tuesday, August 16, 2016

Practical Geometry - as described by those who used it, Part 2

The last post  discussed how Asher Benjamin and Owen Biddle presented Practical Geometry in their pattern books in 1805 and 1806.
This post focuses on Minard Lefever, and finally Peter Nicholson, who inspired them all.

Minard Lefever ( 1798-1854) wrote 5 pattern books between 1829 and 1856.
The Modern Builder's Guide was published in September 1833, in New York.
In his Preface Lefever says " will be proper to specify the authors whom I have either consulted or made extractions from,..."
One of these was Peter Nicholson.  Because Lefever copies Nicholson's drawings  directly I will post only the latter's introductory geometry.

Lefever writes 35 pages of  descriptions for 21 plates on "Geometry Adapted to Practical Carpentry".
Here are Plate 8  and Plate 20.

Minard Lefever, The Modern Builder's Guide, NY, 1833, reprint by Dover Publications, NY, 1969.

Peter Nicholson (1765-1844) practiced architecture, mathematics, and engineering in Scotland.  He taught and wrote 27 books.  The Carpenter's New Guide was first  published in 1792 in Great Britain. His books were regularly reprinted in the States.

The book reproduced here was printed in Philadelphia in 1830, his 10th Edition with, he writes,"6 new Plates".  The book is 121 pages long not including the Index.
27 of those pages are of - as his title page says - Practical Geometry for Carpentry and Joinery, "the whole founded on the geometric principals; the theory and practice well explained and fully exemplified" on 10 copper-plates.

In the Preface he says, " is Geometry which lays down all the first principals of building, measures lines, angles, and solids, and gives rules for describing the various kinds of figures used in buildings; therefore, as a necessary introduction to the art treated of, I have first laid down, and explained in the terms of workmen, such problems of Geometry as are absolutely prerequisite to the well understanding and putting into practice the necessary lines for Carpentry."

His introductory geometry plates match those of Asher Benjamin, Owen Biddle and Minard Lefever, all of whom acknowledge him in their prefaces.

Nicholson's Plate 10 is Lefever's Plate 8.

I will bring this book to the 2016 IPTN Workshops in September. It is fragile.

If you would like to read the titles of Peter Nicholson's books, they are listed at the end of his Wikipedia biography.

Other architectural historians must have looked at the first pages of these books. Everyone cannot have just turned to the illustrations of mantles and window casings, building plans and elevations and ignored the plates on geometry. Why hasn't someone else wondered out loud why so many pages on geometry were included in a book about construction?

Someone must have considered that if Nicholson's The Carpenter's New Guide went through 10 editions and was published in the States as well as Great Britain - as well as being directly copied - that carpenters were reading it, using it, that his information was useful, that maybe we should understand what he wrote.

The builders who came before us used geometry to design and build. The knowledge was taught to the next generation hands-on. Books were not needed.
Boys were 'apprenticed', learned their craft and became 'journeymen', traveling to sites to earn and learn. Eventually these men became full carpenters, 'masters', and were admitted to a guild. The guild system was not always possible in the States. Men quit their apprenticeships. moved west or into cities. The skills and knowledge that masters were expected to impart had to be taught in other ways. Asher Benjamin and others set up a school in Boston. The pattern book was another solution - a way for 'young carpenters'  (to quote Owen Biddle) to teach themselves the necessary construction skills, beginning with geometry.

Monday, August 15, 2016

Practical Geometry - as described by those who used it, Part 1

Asher Benjamin, Owen Biddle, Peter Nicholson, and Minard Lefever

What they wrote about Practical Geometry in their pattern books: Asher Benjamin in 1806, Owen Biddle in 1805, Peter Nicholson beginning in 1792, Minard Lefever in 1833.

I want their words to be easily available to anyone who is curious - someone who comes upon this blog or someone who comes to the 2016 IPTN workshops in September.

Remember that the pictures can be expanded - click on them.

Asher Benjamin's,The American Builder's Companion, was first published in 1806, updated and edited through 6 editions to 1827.

His title included the various chapters he has included. The first is  "Practical Geometry".

In his preface he says, " I have first laid down and explained such problems of Geometry, as are absolutely necessary to the well understanding of the subject."

His first 18 of 114 pages are about using geometry to design and build.

I have copied here his Plate I and its accompanying notes.

Asher Benjamin, The American Builder's Companion, Boston, MA, 6th ( 1827) edition, Dover Publications Reprint, 1969. Benjamin wrote at least 6 pattern books beginning in 1797, all popular.

Owen Biddle's book , Biddle's Young Carpenter's Assistant, 1805, was half the size of Benjamin's, easy to tuck into a tool chest. His first 9 pages of 112 are devoted to Geometry.
First comes how to construct a drafting board and attach paper to it, followed by how to make a T square and what kind of instruments to use. Then he says, " I shall now proceed to explain some of the most useful geometrical problems, which every Carpenter ought to be acquainted with". p.4

Owen Biddle, Biddle's Young Carpenter's Assistant, Philadelphia and New York, 1805, Dover Publications reprint, 2006. This is his only book  A respected master carpenter in Philadelphia, he died in 1806.

to be continued....

Friday, August 12, 2016

2016 Preservation Trades Network Workshops, September 9-11, Clermont Farm, Berryville, Virginia

The annual gathering will be at Clermont  Farm now owned by the Commonwealth of Virginia.

Here is the link to the farm: Their facebook page has good pictures.
The National Barn Alliance will be there too.  On Friday there will be a barn tour -

There will be blacksmiths, lime mortar makers, timber framers, window repair people, masonry specialists, painters, roofers...

 Last year at Shelburne Farm I watched dimensional lumber come out of a log with bark, all by hand. I saw a Georgian cabinet built, and windows become like new.  The pictures are from that gathering.

I will be there to teach 2 sessions on
                Practical Geometry
 which, to quote Owen Biddle in 1805 "every Carpenter ought to be acquainted with".

Or more formally: "Geometry is the foundation on which practical Carpentry is based." Minard Lefever, 1833,

The sessions will be hands-on.
I will have compasses, pencils, erasers and straight edges. And drawings.
I will be helping whoever shows up see the geometry which governed framing and design for churches, mansions, houses, barns. As we uncover the geometry participants will see how design and structure come from the compass.

We will decipher brick houses in Virginia, wood frame churches in New England, houses built from 1680 to 1840. For people who want to see how much they already know I will have the plates from the first pages of the pattern books which present  "such problems in Geometry, as are absolutely necessary to the well understanding of the subject." (Asher Benjamin, 1827) Will they master the problems with a compass and a straight edge?

The pattern books of Asher Benjamin, Owen Biddle, Peter Nicholson, Minard Lefever,  will be available along with posters and handouts on Robert Adam and William Buckland.
And paper for experimenting

I demonstrate twice. There will be  plenty of opportuity for me to watch and learn from the other presenters, to explore the farm and its buildings, and talk with people. I know I will have a great time.

You can come too.

Monday, July 25, 2016

Washington County, NY, House - a Dutch vernacular frame

written January, 2016

The house has been stripped to its frame. The sheathing removed, each board numbered as it came down. The stair and moldings (inside and out) carefully moved into storage.

Now the frame is visible -
no ridge beam,
12 bents: each is a post on either end mortised to a 2nd floor joist.
The plate across the top holds them all together; The 14 rafters sit on the plate and are not spaced to match the bents.

The joists on each end are mortised into posts.
Plates, mortised into the sides of the  posts, space the bents and carry the intermediate 2nd floor joists.

Here is a post with its joist and the pegs that hold the tenons of the plates seen from the outside.

Here seen from the inside, are: 2 posts, an intermediate stud; 2nd floor joists, plates and intermediate joists.

We think it was assembled bent by bent, the intermediate plates added one at a time as each bent was set into the sill.

I have measured the first floor - twice The first time it was just too cold. I hurried. I wasn't careful.

The framer used Hudson Valley Dutch framing. The house was clothed in the latest Federal style with possible Shaker influences. Inside it retains the traditional system.  

I need to add more, especially about the Dutch way of framing.
An orthographic perspective would make the frame easier to read.
The frame details deserve a post of their own.
So does the careful cleaning and repair of the frame by Green Mountain Timber Frames.
I want to redraw the front elevation to reflect the frame we saw and measured compared to the plaster and clapboard surfaces I saw and measured in the beginning.

However, it is July, months later.  Time to share!

Sunday, July 24, 2016

The Old First Church Geometry - the Floor Plan - Part 4

 I first wrote about the geometry of the Old First Church in Bennington, Vermont, in September, 2012, focusing on the 2nd floor windows with their round tops.

I will not repeat that post and the ones that followed -  just expand upon it here.

As I studied how the church was designed I saw that the window design was the logical extension of the basic design.

This spring the full window design and then the geometry of the floor plan - which had eluded me - became obvious.

The circle geometry which determined the curves in the half round top also determined the size of the window itself and muntin pattern  in the lower section.
The completed circle of the top half intersects with the circle which begins in the lower sash. The circles divided in 4 determine the size
of the window panes.

The panes themselves are not quite square because of the thickness of the frame.

The pattern in the rounded top is made by 7 intersecting circles. The window itself is 2 intersecting circles.

I have called these 'rolling circles' because visually they seem able to roll one way or the other. Perhaps in a church the circles roll toward each other and meet..
It would be fitting symbolism for Old First Church whose covenant says the members hope to " ... become a people whom the Lord hath bound up together... "

Here is the floor plan, measured and drawn in the 1930's by Denison Bingham Hull, the architect who supervised the church's restoration.

I superimposed a circle with its rectangle marked in red which  matches the circles that define the east interior elevation and the exterior front elevation.

This is what I had not seen before -  how the geometry of the floor layout uses the same forms as the windows. Both are 2 intersecting circles.

The rectangles laid out by the circles determine the size of the sanctuary. The diamond shape where the 2 circles cross, the center of the church,  is the  location of the dome -an acoustic device - a technological tour-de-force in 1805. The narthex fills and over flows the lower quarter of the circle. The depth and width of the front bay is determined by the arc of the circle's perimeter.

Expanding the circles in the way that the window design    'roll'  I saw that Lavius Fillmore, the master builder, did not need divide his circles into daisy wheels to locate columns and determine proportions.

This relationship of one circle to another in a linear (up and down, side to side) pattern rather than relating one circle to the next by moving around the perimeter is seen in all the elevations and plans for the Old First Church.

In the drawing to the left I have added small circles at the intersections of the arcs which mark the lines of the columns, the corners of the front bay and intersect with the perimeters of the circles at the 4 major columns - the black squares - which run from  piers in the basement through the sanctuary into the attic to anchor the trusses which carry the roof and the trusses from which the dome is suspended.

Fillmore need not have drawn a daisy wheel with its 6 petals to refine his design.
He might just have rolled his circles.

In many ways these different approaches to 'basic geometry' - as Asher Benjamin calls it - cross-reference each other. The daisy wheel and the rolling circles are variations of the same proportions.
My 'aha' moment is when I find for one way of working that is clean, simple and 'obvious'.