PRACTICAL GEOMETRY - Lesson 4

 

The posts in this series  Lessons 1-7  are :

 https://www.jgrarchitect.com/2020/04/lessons.html

 https://www.jgrarchitect.com/2020/04/practical-geometry-lessons-2.html

 https://www.jgrarchitect.com/2020/04/practical-geometry-lesson-3.html

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

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

https://www.jgrarchitect.com/2020/06/practical-geometry-lessons-lesson-5.html

https://www.jgrarchitect.com/2020/06/practical-geometry-lesson-5-addendum.html

https://www.jgrarchitect.com/2020/08/lesson-6-rule-of-thirds-part-1_21.html

https://www.jgrarchitect.com/2020/08/lesson-6-rule-of-thirds-part-2-serlio.html

 

https://www.jgrarchitect.com/2020/09/lesson-7-how-to-layout-frame-with-lines.html

Practical Geometry - what our ancestors called this geometry

Practical Geometry.
It's what our ancestors called these diagrams I draw.


 Here is Peter Nicholson who wrote about Practical Geometry. His writings make clear that geometry was once an expected and necessary part of construction, used both by the designer and the artisan.

His first book, The Carpenter's New Guide, published in London in 1792.

He begins with a Preface, some of which I quoted in an earlier post: http://www.jgrarchitect.com/2016/08/practical-geometry-as-described-by_16.html


Page 2 is copied here.

The use of geometry in construction was so accepted that Peter Nicholson waits until his third paragraph before he shares that geometry is useful in mathematics and science too.



By the time of his death in 1844, Nicholson had published 27 books in London, New York City, and Philadelphia.  More than 10 years later his books were still in print.

This portrait is in his updated book The New and Improved Practical Builder, published in 1837.

This time he writes a whole paragraph explaining Practical Geometry. Has he been asked to be more thorough? Have the new uses of geometry in science changed the perception of what geometry is? Have men become carpenters by necessity - especially in the New World - rather than by apprenticeship, and thus desire to educate themselves?




Here is the Introductory Chapter. 

The second paragraph describes the 2 branches of Geometry: Theoretical and Practical.
Now  the Theory of  Geometry is carefully described, including a reference to Euclid, but it is still one of the 2 branches of Geometry.
The other branch, Practical Geometry,
 allows "the architect to regulate his designs and the artisan to construct his lines".



Later, on page vii, he writes, "There is no mechanical profession that does not derive considerable advantage from it."


first portrait: by James Green, 1816, now in the National Portrait Gallery, London
second portrait: the frontispiece of The New and Improved Practical Builder. Don't miss his compass.

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 "...it 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, "...it 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.

Part 1 can be read here: http://www.jgrarchitect.com/2016/08/practical-geometry-as-described-by.html

Learning from a Workshop

The IPN Workshops at the Shelburne Farms Coach Barn were superb.
The barn is magnificent. To be able to be in and around it for 4 days was true luxury.

Above left is the main entrance from the court yard. Right is the dormer for the hay loft door above the stable. Below is half of one barn door showing its hinges and brick work.



The food was plentiful and excellent - local and fresh.
The company and the workshops couldn't be beat.
Of course I plan to go to the 20th annual IPT Workshop to be held in Virginia, autumn of 2016 . www.ptn.org


My presentation was almost derailed by the cheap school compasses I brought. The compasses did not hold their angle, so the diagrams we drew weren't true. I had not anticipated that the participants might not know how to draft: they needed basic instructions and better tools.

Luckily people bore with me and I presented twice. Many people talked with me about geometry between sessions.




Here is what worked best:

The daisy wheel: As people found the rectangle created by 4 points  they easily understood the geometry of the Old First Church in Bennington.








Making a square:  beginning with a line and a circle.

The hardest part for people to figure out was how to draw the arcs for the vertical line. I felt very successful when I heard one person explaining how to do it to another.










The 1830's farmer's cottage pleased everyone. They could see how to use what they had drawn.


A few people were able to rotate the square 45* to complete the diagram as shown

I brought the pictures from my post on  Asher Benjamin, Owen Biddle and Peter Nicholson. http://www.jgrarchitect.com/2015/07/geometry-of-cobb-hepburn-house-part-2.html

 We were to draw squares based on their diagrams  - as shown here:
This is where, especially, the compasses were not up to the task. The squares were not true;they were cock-eyed. I loaned my good compass out - so much easier to draw with good tools!
So, I explained and demonstrated. People practiced.
The pattern books' first pages of geometry turned out to be an adventure.


I thought to show 4 different ways to grow a layout from one dimension.
Instead I used the different buildings and diagrams as illustrations as people asked questions,
Good, thoughtful questions.

What a good time we had!

A barn built in the 1830's

Green Mountain Timber Frames http://www.greenmountaintimberframes.com/ measured this barn before they dismantled it to use its frame anew.

Due to the wood used - poplar, beech, hemlock - the layout and the construction we think this barn was built by a farmer without an extensive background in framing. We think it dates to the  1830's.

 The floor was dirt, the head room under the hay loft not quite 6 ft.
What was it used for? Sheep perhaps? Sheds, windows and a silo were added over the last 180 years, making the original purpose hard to read.


I start with the farmer.
He had some wood of a certain size and length he could use for posts and beams for a barn. He knew how the barn would be used and where it would go.

Probably he had a carpenter square - they were readily available. But maybe not, as his dimensions don't quite fit. And he was much more comfortable with the old-fashioned geometry of the 'whirling square'.

He started with the width - 18 feet. He made a square: 18 ft wide, 18 ft. high - first diagram.\
Or so it seems. Today that height is 17'-10", 2". I think originally the width was also 17'-10". His inch seems to have been just a bit smaller than today's inch

He could have started with a string about 18 ft. long. He could have used a compass with a 27" radius, stepped it out twice for 4'-6", twice more for 9' and doubled that for 18'; or a pole 4'-6" long.

In his square he laid out his center lines and then the star that joins the points - the second and third diagrams. This is a medieval framing system which came to New England with the English colonists.

I have added circles to mark how the lines of the star cross at the locations of the girts, I've added a green dashed line to show how the height of the wall is 2/3 the height of the end wall. Almost. It's off by 2"

Using a carpenter square to layout a 3/4/5 triangle does not work as well. The wall height isn't high enough. The lower girt can be determined - see the green circles - but not the upper one.

While the frame appears governed by the traditional English framing geometry, the frame itself has dropped girts - a Dutch traditional way of framing. The girts are mortised into the posts below the upper beam. This combination of framing methods is sometimes referred to as 'American'.


The floor plan is simple: three 3/4/5 triangles. If the width is 18'  the length should be 40'-6" . It was 40'-2" measured on site. The men repairing the frame tell me it is 40'-1"; that the 2 interior bents are at 13'-4 1/2" from each end.

If one arm of the 3/4/5 triangle is 17'-10", the other is 13'-4 1/2".
3/4 of one side of a 17'-10" square = 13'-4 1/2". So either framing system fits the floor plan.

Dan McKeen, GM Timber Frames, also tell me 3 girts are beech, one poplar.The top plates are poplar and in good shape. The posts are sawn hemlock and hewn beech. The ties are sawn hemlock.

I looked at how did this farmer/framer laid out his girts in the side walls.
Here I tried - on the right in red - 3/4/5 triangles. The intersections - red circles - using a triangle that includes the rafter tails, are close, but convoluted. Not simple.
However, a square laid out inside the frame - on the left in green - neatly divides the space in thirds - green circles.

The star of the square used for the gable end laid out along the side also notes the placement of all the  girts.








The numbers on the early carpenter squares were engraved by hand. It is possible that this farmer/framer owned a square that was very slightly off. Or he used his own measure.

The man who built this frame comes alive as I study it; I've met him. Now I want to ask how he learned to frame - who taught him? what tools did he like? where did he start? were we right about his choice of materials?

A Circle to a Square, 1830's cottage north of Boston, Part 3

 When I was preparing for my presentation at the Timber Framers Guild annual conference in August I assumed people knew the basic geometry - how to begin with a dot and a line and construct a shape - for example, a square - that was useful in construction. I found that I was wrong.




Here are diagrams showing how to lay out a square, beginning with a line, a dot and then: another dot.






The two dots define a distance, a radius.  So a circle is drawn. Now there are 3 dots on the line, all related to each other.






 Draw 2 more circles using the dots on the circumference of the circle as the centers. All the circles are the same size.

Then take distance, whatever length you like - but it should fit on your page! Swing 2 arcs using that length and the centers of your two outer circles, one above and one below the first circle. The arcs cross.
Draw a line between the points where they cross. This line is perpendicular to your first line. and it crosses the first circle at 2 points.


Add 2 more circles, above and below the first row of 3, using the   points you just found as the centers. Now you have 5 circles all the same size, placed in relationship to each other. And you have 4 petals made by the arcs of the circles. Join the outside points of the petals, also where the circles cross. This is a square - drawn here in red.

Of course, your length (radius) needs to be constant...You need to be draft carefully. This is precision work.




This diagram applies to the 1830 house in Reading MA, which I have written about earlier.
While preparing for the Guild presentation, I realized how more complex and also how much simpler the diagram could be.

Here is the 'square construction' laid over the elevation, the base of the square on the first floor. The center line defines the ceiling of the fisrt floor, the placement of the second floor joist. The top of the square defines the ceiling of the 2nd floor, the placement of the ceiling joists which are also collar ties.
I superimposed a square diamond on the diagram - in green - because the intersections of the diamond and the square's petals marked the windows' width.





Then, I tried turning the square 45*. The upper sides of the new square is the pitch and location of the roof rafters.   The intersection of the 2 squares is the top plate of the wall, the placement for the roof rafters. The petals are the width of the windows.



All this became visible, obvious, while  I was working intently on preparing a coherent talk. I learned a lot. Now I am beginning to re-post and update my earlier diagrams.

Here is the farm house that inspired my original research. The first floor parlors had  such presence and grace I can feel them today.
The left hand wing is a later addition.









for Part 1 see: http://www.jgrarchitect.com/2010/08/regulating-lines-1830s-cottage.html
for Part 2 see: http://www.jgrarchitect.com/2013/05/regulating-lines-1830s-house-north-of.html

1830's cottage north of Boston, Part 2

This is an update to the post about the floor plan of this house written in August, 2010.

 Last week I realized that the builder of this cottage would probably have used the same kind of geometry for the elevations that he used for  the floor plan.

This is an obvious observation. Really! Makes me roll my eyes to know that it only took me 3 years to figure this out!
It is clear from the photograph that the house has been expanded over the years. The siding was 'updated' about 40 years ago. My diagrams refer only to the part of the house under the gable, to the right.




So here are the diagrams and drawings - click to enlarge:

The floor plan is shown with the complete daisy wheel for reference. The main rectangle of the house plan is determined by where 4 petals of the daisy touch the circle; the wing's size determined by the arcs of 2 of the outer circles. This is a straight forward use of circle geometry. The elevations use a slightly different pattern.





The side elevation measure by me and drawn to scale:

.
The circle geometry I think the builder used for the elevations:
The upper diagram shows 2 circles intersecting - the outside of each touches the center of the other. The place where the circles overlap is called a vesica piscis ('fish bladder' in Latin). The red line bisects the vesica piscis.
The lower diagram shows one circle with 4 circles intersecting it, creating 4 intersecting vesicae piscis. The red lines are only partially shown for clarity.
The diagrams show how the shape and dimensions of the elevation were determined.

On the elevation the circle is the same size as that used for the floor plan. Its radius is the length of the wall. It surrounds the house. Its center is the fat red line between the second floor windows. The upper circle is drawn in full, but only part of the lower circle is shown. The lines through the vesicae piscis determine the peak of the roof, the center of the first floor windows.
I have drawn only part of one of the circles on the sides, the left one. The vesica piscis there follows  the wall of the house.
 Note that the center of the main circle not only positions the 2nd floor windows but also marks the square of the wall, the top of which is where the builder will set his rafters.The square is outlined in green. The circle which determined the floor plan, its dimensions, is also used to determine dimensions of the elevations.


for part 1 see: http://www.jgrarchitect.com/2010/08/regulating-lines-1830s-cottage.html
for Part 3 see: http://www.jgrarchitect.com/2014/09/how-to-construct-square.html

1830's cottage north of Boston, Part 1

----

A family with a small c. 1830 story-and-a-half house asked me for advice - how to expand their home in keeping with its character.

We talked about their needs, the house constraints, the land. I suggested that a variation of "big house, little house, back house, barn" might be work. (The old jump rope rhyme refers to the massing of early New England farms. There is a book of that name describing the phenomenon.)
Back on my drafting board - after I had drawn up a sketch for the family and their contractor which reflected our discussion - I played a bit with the form. Here's what came out of it:


The floor plan provided to me was at 1/8" = 1'-0", very
small. However I later measured the house. The plan is accurate. See my post in May, 2013, for the elevations. I am happy to discuss the date of the house, if asked.

The first floor plan and a sketch of the house is here. The red line shows the outline of the original c. 1830 house plan.

The second drawing shows the original floor plan with - in red - the length of the side used as a radius for a circle.


The third drawing shows that a circle whose radius is the length of the
wall touches the 4 corners of the main house.The rectangle of the house
is determined by the circle. An arc using the same radius defines the
length of the wing, but not quite the width. The interior arcs of the
daisy wheel also define the small front entry. The 6 points of the 'daisy wheel' are shown, only 2 of the 6 'daisy petals' are drawn.

The house is simply built, perhaps by a farmer, not a joiner. Whoever he was, whatever education he had in construction, he knew circle geometry and used it to layout this house.